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1.10: Partial Fractions

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Partial fractions is the name given to a technique of integration that may be used to integrate any rational function 1. We already know how to integrate some simple rational functions

1xdx=log|x|+C11+x2dx=arctan(x)+C

Combining these with the substitution rule, we can integrate similar but more complicated rational functions:

12x+3dx=12log|2x+3|+C13+4x2dx=123arctan(2x3)+C

By summing such terms together we can integrate yet more complicated forms

[x+1x+1+1x1]dx=x22+log|x+1|+log|x1|+C

However we are not (typically) presented with a rational function nicely decomposed into neat little pieces. It is far more likely that the rational function will be written as the ratio of two polynomials. For example:

x3+xx21dx

In this specific example it is not hard to confirm that

x+1x+1+1x1=x(x+1)(x1)+(x1)+(x+1)(x+1)(x1)=x3+xx21

and hence

x3+xx21dx=[x+1x+1+1x1]dx=x22+log|x+1|+log|x1|+C

Of course going in this direction (from a sum of terms to a single rational function) is straightforward. To be useful we need to understand how to do this in reverse: decompose a given rational function into a sum of simpler pieces that we can integrate.

Suppose that N(x) and D(x) are polynomials. The basic strategy is to write N(x)D(x) as a sum of very simple, easy to integrate rational functions, namely

  1. polynomials — we shall see below that these are needed when the degree 2 of N(x) is equal to or strictly bigger than the degree of D(x), and
  2. rational functions of the particularly simple form A(ax+b)n and
  3. rational functions of the form Ax+B(ax2+bx+c)m.

We already know how to integrate the first two forms, and we'll see how to integrate the third form in the near future.

To begin to explore this method of decomposition, let us go back to the example we just saw

x+1x+1+1x1=x(x+1)(x1)+(x1)+(x+1)(x+1)(x1)=x3+xx21

The technique that we will use is based on two observations:

  1. The denominators on the left-hand side of are the factors of the denominator x21=(x1)(x+1) on the right-hand side.
  2. Use P(x) to denote the polynomial on the left hand side, and then use N(x) and D(x) to denote the numerator and denominator of the right hand side. That is

    P(x)=xN(x)=x3+xD(x)=x21.

    Then the degree of N(x) is the sum of the degrees of P(x) and D(x). This is because the highest degree term in N(x) is x3, which comes from multiplying P(x) by D(x), as we see in

    x+1x+1+1x1=P(x)xD(x)(x+1)(x1)+(x1)+(x+1)(x+1)(x1)=x3+xx21

    More generally, the presence of a polynomial on the left hand side is signalled on the right hand side by the fact that the degree of the numerator is at least as large as the degree of the denominator.

Partial fraction decomposition examples

Rather than writing up the technique — known as the partial fraction decomposition — in full generality, we will instead illustrate it through a sequence of examples.

Example 1.10.1 x3x23x+2dx

In this example, we integrate N(x)D(x)=x3x23x+2.

Solution

  • Step 1. We first check to see if a polynomial P(x) is needed. To do so, we check to see if the degree of the numerator, N(x), is strictly smaller than the degree of the denominator D(x). In this example, the numerator, x3, has degree one and that is indeed strictly smaller than the degree of the denominator, x23x+2, which is two. In this case 3 we do not need to extract a polynomial P(x) and we move on to step 2.
  • Step 2. The second step is to factor the denominator

    x23x+2=(x1)(x2)

    In this example it is quite easy, but in future examples (and quite possibly in your homework, quizzes and exam) you will have to work harder to factor the denominator. In Appendix A.16 we have written up some simple tricks for factoring polynomials. We will illustrate them in Example 1.10.3 below.
  • Step 3. The third step is to write x3x23x+2 in the form

    x3x23x+2=Ax1+Bx2

    for some constants A and B. More generally, if the denominator consists of n different linear factors, then we decompose the ratio as

    rational function=A1linear factor 1+A2linear factor 2++Anlinear factor n

    To proceed we need to determine the values of the constants A, B and there are several different methods to do so. Here are two methods

  • Step 3 — Algebra Method. This approach has the benefit of being conceptually clearer and easier, but the downside is that it is more tedious.

    To determine the values of the constants A, B, we put 4 the right-hand side back over the common denominator (x1)(x2).

    x3x23x+2=Ax1+Bx2=A(x2)+B(x1)(x1)(x2)

    The fraction on the far left is the same as the fraction on the far right if and only if their numerators are the same.

    x3=A(x2)+B(x1)

    Write the right hand side as a polynomial in standard form (i.e. collect up all x terms and all constant terms)

    x3=(A+B)x+(2AB)

    For these two polynomials to be the same, the coefficient of x on the left hand side and the coefficient of x on the right hand side must be the same. Similarly the coefficients of x0 (i.e. the constant terms) must match. This gives us a system of two equations.

    A+B=12AB=3

    in the two unknowns A,B. We can solve this system by

    • using the first equation, namely A+B=1, to determine A in terms of B:

      A=1B

    • Substituting this into the remaining equation eliminates the A from second equation, leaving one equation in the one unknown B, which can then be solved for B:

      2AB=3substitute A=1B2(1B)B=3clean up2+B=3so B=1

    • Once we know B, we can substitute it back into A=1B to get A.

      A=1B=1(1)=2

    Hence

    x3x23x+2=2x11x2

  • Step 3 — Sneaky Method. This takes a little more work to understand, but it is more efficient than the algebra method.

    We wish to find A and B for which

    x3(x1)(x2)=Ax1+Bx2

    Note that the denominator on the left hand side has been written in factored form.

    • To determine A, we multiply both sides of the equation by A's denominator, which is x1,

      x3x2=A+(x1)Bx2

      and then we completely eliminate B from the equation by evaluating at x=1. This value of x is chosen to make x1=0.

      x3x2|x=1=A+(x1)Bx2|x=1A=1312=2

    • To determine B, we multiply both sides of the equation by B's denominator, which is x2,

      x3x1=(x2)Ax1+B

      and then we completely eliminate A from the equation by evaluating at x=2. This value of x is chosen to make x2=0.

      x3x1|x=2=(x2)Ax1|x=2+BB=2321=1

    Hence we have (the thankfully consistent answer)

    x3x23x+2=2x11x2

    Notice that no matter which method we use to find the constants we can easily check our answer by summing the terms back together:

    2x11x2=2(x2)(x1)(x2)(x1)=2x4x+1x23x+2=x3x23x+2

  • Step 4. The final step is to integrate.

    x3x23x+2dx=2x1dx+1x2dx=2log|x1|log|x2|+C

Perhaps the first thing that you notice is that this process takes quite a few steps 5. However no single step is all that complicated; it only takes practice. With that said, let's do another, slightly more complicated, one.

Example 1.10.2 3x38x2+4x1x23x+2dx

In this example, we integrate N(x)D(x)=3x38x2+4x1x23x+2.

Solution

  • Step 1. We first check to see if the degree of the numerator N(x) is strictly smaller than the degree of the denominator D(x). In this example, the numerator, 3x38x2+4x1, has degree three and the denominator, x23x+2, has degree two. As 32, we have to implement the first step.

    The goal of the first step is to write N(x)D(x) in the form

    N(x)D(x)=P(x)+R(x)D(x)

    with P(x) being a polynomial and R(x) being a polynomial of degree strictly smaller than the degree of D(x). The right hand side is P(x)D(x)+R(x)D(x), so we have to express the numerator in the form N(x)=P(x)D(x)+R(x), with P(x) and R(x) being polynomials and with the degree of R being strictly smaller than the degree of D. P(x)D(x) is a sum of expressions of the form axnD(x). We want to pull as many expressions of this form as possible out of the numerator N(x), leaving only a low degree remainder R(x).

    We do this using long division — the same long division you learned in school, but with the base 10 replaced by x.

    • We start by observing that to get from the highest degree term in the denominator (x2) to the highest degree term in the numerator (3x3), we have to multiply it by 3x. So we write,

      In the above expression, the denominator is on the left, the numerator is on the right and 3x is written above the highest order term of the numerator. Always put lower powers of x to the right of higher powers of x — this mirrors how you do long division with numbers; lower powers of ten sit to the right of higher powers of ten.

    • Now we subtract 3x times the denominator, x23x+2, which is 3x39x2+6x, from the numerator.
    • This has left a remainder of x22x1. To get from the highest degree term in the denominator (x2) to the highest degree term in the remainder (x2), we have to multiply by 1. So we write,
    • Now we subtract 1 times the denominator, x23x+2, which is x23x+2, from the remainder.
    • This leaves a remainder of x3. Because the remainder has degree 1, which is smaller than the degree of the denominator (being degree 2), we stop.
    • In this example, when we subtracted 3x(x23x+2) and 1(x23x+2) from 3x38x2+4x1 we ended up with x3. That is,

      3x38x2+4x1  3x(x23x+2)  1(x23x+2)=x3

      or, collecting the two terms proportional to (x23x+2)

      3x38x2+4x1  (3x+1)(x23x+2) = x3

      Moving the (3x+1)(x23x+2) to the right hand side and dividing the whole equation by x23x+2 gives

      3x38x2+4x1x23x+2 = 3x+1 + x3x23x+2

      And we can easily check this expression just by summing the two terms on the right-hand side.

    We have written the integrand in the form N(x)D(x)=P(x)+R(x)D(x), with the degree of R(x) strictly smaller than the degree of D(x), which is what we wanted. Observe that R(x) is the final remainder of the long division procedure and P(x) is at the top of the long division computation. This is the end of Step 1. Oof! You should definitely practice this step.

  • Step 2. The second step is to factor the denominator

    x23x+2=(x1)(x2)

    We already did this in Example 1.10.1.
  • Step 3. The third step is to write x3x23x+2 in the form

    x3x23x+2=Ax1+Bx2

    for some constants A and B. We already did this in Example 1.10.1. We found A=2 and B=1.
  • Step 4. The final step is to integrate.

    3x38x2+4x1x23x+2dx=[3x+1]dx+2x1dx+1x2dx=32x2+x+2log|x1|log|x2|+C

You can see that the integration step is quite quick — almost all the work is in preparing the integrand.

Here is a very solid example. It is quite long and the steps are involved. However please persist. No single step is too difficult.

Example 1.10.3 x4+5x3+16x2+26x+22x3+3x2+7x+5dx

In this example, we integrate N(x)D(x)=x4+5x3+16x2+26x+22x3+3x2+7x+5.

Solution

  • Step 1. Again, we start by comparing the degrees of the numerator and denominator. In this example, the numerator, x4+5x3+16x2+26x+22, has degree four and the denominator, x3+3x2+7x+5, has degree three. As 43, we must execute the first step, which is to write N(x)D(x) in the form

    N(x)D(x)=P(x)+R(x)D(x)

    with P(x) being a polynomial and R(x) being a polynomial of degree strictly smaller than the degree of D(x). This step is accomplished by long division, just as we did in Example 1.10.3. We'll go through the whole process in detail again.

    Actually — before you read on ahead, please have a go at the long division. It is good practice.

    • We start by observing that to get from the highest degree term in the denominator (x3) to the highest degree term in the numerator (x4), we have to multiply by x. So we write,
    • Now we subtract x times the denominator x3+3x2+7x+5, which is x4+3x3+7x2+5x, from the numerator.
    • The remainder was 2x3+9x2+21x+22. To get from the highest degree term in the denominator (x3) to the highest degree term in the remainder (2x3), we have to multiply by 2. So we write,
    • Now we subtract 2 times the denominator x3+3x2+7x+5, which is 2x3+6x2+14x+10, from the remainder.
    • This leaves a remainder of 3x2+7x+12. Because the remainder has degree 2, which is smaller than the degree of the denominator, which is 3, we stop.
    • In this example, when we subtracted x(x3+3x2+7x+5) and 2(x3+3x2+7x+5) from x4+5x3+16x2+26x+22 we ended up with 3x2+7x+12. That is,

      x4+5x3+16x2+26x+22  x(x3+3x2+7x+5)  2(x3+3x2+7x+5)=3x2+7x+12

      or, collecting the two terms proportional to (x3+3x2+7x+5) we get

      x4+5x3+16x2+26x+22  (x+2)(x3+3x2+7x+5)= 3x2+7x+12

      Moving the (x+2)(x3+3x2+7x+5) to the right hand side and dividing the whole equation by x3+3x2+7x+5 gives

      x4+5x3+16x2+26x+22x3+3x2+7x+5=x+2+3x2+7x+12x3+3x2+7x+5

    This is of the form N(x)D(x)=P(x)+R(x)D(x), with the degree of R(x) strictly smaller than the degree of D(x), which is what we wanted. Observe, once again, that R(x) is the final remainder of the long division procedure and P(x) is at the top of the long division computation.

  • Step 2. The second step is to factor the denominator D(x)=x3+3x2+7x+5. In the “real world” factorisation of polynomials is often very hard. Fortunately 6, this is not the “real world” and there is a trick available to help us find this factorisation. The reader should take some time to look at Appendix A.16 before proceeding.
    • The trick exploits the fact that most polynomials that appear in homework assignments and on tests have integer coefficients and some integer roots. Any integer root of a polynomial that has integer coefficients, like D(x)=x3+3x2+7x+5, must divide the constant term of the polynomial exactly. Why this is true is explained 7 in Appendix A.16.
    • So any integer root of x3+3x2+7x+5 must divide 5 exactly. Thus the only integers which can be roots of D(x) are ±1 and ±5. Of course, not all of these give roots of the polynomial — in fact there is no guarantee that any of them will be. We have to test each one.
    • To test if +1 is a root, we sub x=1 into D(x):

      D(1)=13+3(1)2+7(1)+5=16

      As D(1)0, 1 is not a root of D(x).
    • To test if 1 is a root, we sub it into D(x):

      D(1)=(1)3+3(1)2+7(1)+5=1+37+5=0

      As D(1)=0, 1 is a root of D(x). As 1 is a root of D(x), (x(1))=(x+1) must factor D(x) exactly. We can factor the (x+1) out of D(x)=x3+3x2+7x+5 by long division once again.
    • Dividing D(x) by (x+1) gives:

      This time, when we subtracted x2(x+1) and 2x(x+1) and 5(x+1) from x3+3x2+7x+5 we ended up with 0 — as we knew would happen, because we knew that x+1 divides x3+3x2+7x+5 exactly. Hence

      x3+3x2+7x+5  x2(x+1)  2x(x+1)  5(x+1) = 0

      or

      x3+3x2+7x+5 = x2(x+1) + 2x(x+1) + 5(x+1)

      or

      x3+3x2+7x+5=(x2+2x+5)(x+1)

    • It isn't quite time to stop yet; we should attempt to factor the quadratic factor, x2+2x+5. We can use the quadratic formula 8 to find the roots of x2+2x+5:

      b±b24ac2a=2±4202=2±162

      Since this expression contains the square root of a negative number the equation x2+2x+5=0 has no real solutions; without the use of complex numbers, x2+2x+5 cannot be factored.

    We have reached the end of step 2. At this point we have

    x4+5x3+16x2+26x+22x3+3x2+7x+5=x+2+3x2+7x+12(x+1)(x2+2x+5)

  • Step 3. The third step is to write 3x2+7x+12(x+1)(x2+2x+5) in the form

    3x2+7x+12(x+1)(x2+2x+5)=Ax+1+Bx+Cx2+2x+5

    for some constants A, B and C.

    Note that the numerator, Bx+C of the second term on the right hand side is not just a constant. It is of degree one, which is exactly one smaller than the degree of the denominator, x2+2x+5. More generally, if the denominator consists of n different linear factors and m different quadratic factors, then we decompose the ratio as

    rational function=A1linear factor 1+A2linear factor 2++Anlinear factor n=+B1x+C1quadratic factor 1+B2x+C2quadratic factor 2++Bmx+Cmquadratic factor m

    To determine the values of the constants A, B, C, we put the right hand side back over the common denominator (x+1)(x2+2x+5).

    3x2+7x+12(x+1)(x2+2x+5)=Ax+1+Bx+Cx2+2x+5=A(x2+2x+5)+(Bx+C)(x+1)(x+1)(x2+2x+5)

    The fraction on the far left is the same as the fraction on the far right if and only if their numerators are the same.

    3x2+7x+12=A(x2+2x+5)+(Bx+C)(x+1)

    Again, as in Example 1.10.1, there are a couple of different ways to determine the values of A, B and C from this equation.

  • Step 3 — Algebra Method. The conceptually clearest procedure is to write the right hand side as a polynomial in standard form (i.e. collect up all x2 terms, all x terms and all constant terms)

    3x2+7x+12=(A+B)x2+(2A+B+C)x+(5A+C)

    For these two polynomials to be the same, the coefficient of x2 on the left hand side and the coefficient of x2 on the right hand side must be the same. Similarly the coefficients of x1 must match and the coefficients of x0 must match.

    This gives us a system of three equations

    A+B=32A+B+C=75A+C=12

    in the three unknowns A,B,C. We can solve this system by

    • using the first equation, namely A+B=3, to determine A in terms of B:   A=3B.
    • Substituting A=3B into the remaining two equations eliminates the A's from these two equations, leaving two equations in the two unknowns B and C.

      A=3B2A+B+C=75A+C=122(3B)+B+C=75(3B)+C=12B+C=15B+C=3

    • Now we can use the equation B+C=1, to determine B in terms of C: B=C1.
    • Substituting this into the remaining equation eliminates the B's leaving an equation in the one unknown C, which is easy to solve.

      B=C15B+C=35(C1)+C=34C=8

    • So C=2, and then B=C1=1, and then A=3B=2. Hence

      3x2+7x+12(x+1)(x2+2x+5)=2x+1+x+2x2+2x+5

  • Step 3 — Sneaky Method. While the above method is transparent, it is rather tedious. It is arguably better to use the second, sneakier and more efficient, procedure. In order for

    3x2+7x+12=A(x2+2x+5)+(Bx+C)(x+1)

    the equation must hold for all values of x.

    • In particular, it must be true for x=1. When x=1, the factor (x+1) multiplying Bx+C is exactly zero. So B and C disappear from the equation, leaving us with an easy equation to solve for A:

      3x2+7x+12|x=1=[A(x2+2x+5)+(Bx+C)(x+1)]x=18=4AA=2

    • Sub this value of A back in and simplify.

      3x2+7x+12=2(x2+2x+5)+(Bx+C)(x+1)x2+3x+2=(Bx+C)(x+1)

      Since (x+1) is a factor on the right hand side, it must also be a factor on the left hand side.

      (x+2)(x+1)=(Bx+C)(x+1)(x+2)=(Bx+C)B=1, C=2

    So again we find that

    3x2+7x+12(x+1)(x2+2x+5)=2x+1+x+2x2+2x+5

    Thus our integrand can be written as

    x4+5x3+16x2+26x+22x3+3x2+7x+5=x+2+2x+1+x+2x2+2x+5.

  • Step 4. Now we can finally integrate! The first two pieces are easy.

    (x+2)dx=12x2+2x2x+1dx=2log|x+1|

    (We're leaving the arbitrary constant to the end of the computation.)

    The final piece is a little harder. The idea is to complete the square 9 in the denominator

    x+2x2+2x+5=x+2(x+1)2+4

    and then make a change of variables to make the fraction look like ay+by2+1. In this case

    x+2(x+1)2+4=14x+2(x+12)2+1

    so we make the change of variables y=x+12,dy=dx2, x=2y1,dx=2dy

    x+2(x+1)2+4dx=14x+2(x+12)2+1dx=14(2y1)+2y2+12dy = 122y+1y2+1dy=yy2+1dy+121y2+1dy

    Both integrals are easily evaluated, using the substitution u=y2+1, du=2ydy for the first.

    yy2+1dy = 1udu2 = 12log|u|=12log(y2+1)= 12log[(x+12)2+1]121y2+1dy = 12arctany = 12arctan(x+12)

    That's finally it. Putting all of the pieces together

    x4+5x3+16x2+26x+22x3+3x2+7x+5dx=12x2+2x+2log|x+1|+12log[(x+12)2+1]+12arctan(x+12)+C

The best thing after working through a few a nice long examples is to do another nice long example — it is excellent practice 10. We recommend that the reader attempt the problem before reading through our solution.

Example 1.10.4 4x3+23x2+45x+27x3+5x2+8x+4dx

In this example, we integrate N(x)D(x)=4x3+23x2+45x+27x3+5x2+8x+4.

  • Step 1. The degree of the numerator N(x) is equal to the degree of the denominator D(x), so the first step to write N(x)D(x) in the form

    N(x)D(x)=P(x)+R(x)D(x)

    with P(x) being a polynomial (which should be of degree 0, i.e. just a constant) and R(x) being a polynomial of degree strictly smaller than the degree of D(x). By long division

    so

    4x3+23x2+45x+27x3+5x2+8x+4=4+3x2+13x+11x3+5x2+8x+4

  • Step 2. The second step is to factorise D(x)=x3+5x2+8x+4.
    • To start, we'll try and guess an integer root. Any integer root of D(x) must divide the constant term, 4, exactly. Only ±1, ±2, ±4 can be integer roots of x3+5x2+8x+4.
    • We test to see if ±1 are roots.

      D(1)=(1)3+5(1)2+8(1)+40 x=1 is not a rootD(1)=(1)3+5(1)2+8(1)+4=0 x=1 is a root

      So (x+1) must divide x3+5x2+8x+4 exactly.
    • By long division

      so

      x3+5x2+8x+4=(x+1)(x2+4x+4)=(x+1)(x+2)(x+2)

    • Notice that we could have instead checked whether or not ±2 are roots

      D(2)=(2)3+5(2)2+8(2)+40 x=2 is not a rootD(2)=(2)3+5(2)2+8(2)+4=0 x=2 is a root

      We now know that both 1 and 2 are roots of x3+5x2+8x+4 and hence both (x+1) and (x+2) are factors of x3+5x2+8x+4. Because x3+5x2+8x+4 is of degree three and the coefficient of x3 is 1, we must have x3+5x2+8x+4=(x+1)(x+2)(x+a) for some constant a. Multiplying out the right hand side shows that the constant term is 2a. So 2a=4 and a=2.

    This is the end of step 2. We now know that

    4x3+23x2+45x+27x3+5x2+8x+4 = 4+3x2+13x+11(x+1)(x+2)2

  • Step 3. The third step is to write 3x2+13x+11(x+1)(x+2)2 in the form

    3x2+13x+11(x+1)(x+2)2=Ax+1+Bx+2+C(x+2)2

    for some constants A, B and C.

    Note that there are two terms on the right hand arising from the factor (x+2)2. One has denominator (x+2) and one has denominator (x+2)2. More generally, for each factor (x+a)n in the denominator of the rational function on the left hand side, we include

    A1x+a+A2(x+a)2++An(x+a)n

    in the partial fraction decomposition on the right hand side 11.

    To determine the values of the constants A, B, C, we put the right hand side back over the common denominator (x+1)(x+2)2.

    3x2+13x+11(x+1)(x+2)2=Ax+1+Bx+2+C(x+2)2=A(x+2)2+B(x+1)(x+2)+C(x+1)(x+1)(x+2)2

    The fraction on the far left is the same as the fraction on the far right if and only if their numerators are the same.

    3x2+13x+11=A(x+2)2+B(x+1)(x+2)+C(x+1)

    As in the previous examples, there are a couple of different ways to determine the values of A, B and C from this equation.

  • Step 3 — Algebra Method. The conceptually clearest procedure is to write the right hand side as a polynomial in standard form (i.e. collect up all x2 terms, all x terms and all constant terms)

    3x2+13x+11=(A+B)x2+(4A+3B+C)x+(4A+2B+C)

    For these two polynomials to be the same, the coefficient of x2 on the left hand side and the coefficient of x2 on the right hand side must be the same. Similarly the coefficients of x1 and the coefficients of x0 (i.e. the constant terms) must match. This gives us a system of three equations,

    A+B=34A+3B+C=134A+2B+C=11

    in the three unknowns A,B,C. We can solve this system by

    • using the first equation, namely A+B=3, to determine A in terms of B:   A=3B.
    • Substituting this into the remaining equations eliminates the A, leaving two equations in the two unknown B,C.

      4(3B)+3B+C=134(3B)+2B+C=11

      or

      B+C=12B+C=1

    • We can now solve the first of these equations, namely B+C=1, for B in terms of C, giving B=C1.
    • Substituting this into the last equation, namely 2B+C=1, gives 2(C1)+C=1 which is easily solved to give
    • C=3, and then B=C1=2 and then A=3B=1.

    Hence

    4x3+23x2+45x+27x3+5x2+8x+4=4+3x2+13x+11(x+1)(x+2)2=4+1x+1+2x+2+3(x+2)2

  • Step 3 — Sneaky Method. The second, sneakier, method for finding A, B and C exploits the fact that 3x2+13x+11=A(x+2)2+B(x+1)(x+2)+C(x+1) must be true for all values of x. In particular, it must be true for x=1. When x=1, the factor (x+1) multiplying B and C is exactly zero. So B and C disappear from the equation, leaving us with an easy equation to solve for A:

    3x2+13x+11|x=1=[A(x+2)2+B(x+1)(x+2)+C(x+1)]x=11=A

    Sub this value of A back in and simplify.

    3x2+13x+11=(1)(x+2)2+B(x+1)(x+2)+C(x+1)2x2+9x+7=B(x+1)(x+2)+C(x+1)=(xB+2B+C)(x+1)

    Since (x+1) is a factor on the right hand side, it must also be a factor on the left hand side.

    (2x+7)(x+1)=(xB+2B+C)(x+1)(2x+7)=(xB+2B+C)

    For the coefficients of x to match, B must be 2. For the constant terms to match, 2B+C must be 7, so C must be 3. Hence we again have

    4x3+23x2+45x+27x3+5x2+8x+4=4+3x2+13x+11(x+1)(x+2)2=4+1x+1+2x+2+3(x+2)2

  • Step 4. The final step is to integrate

    4x3+23x2+45x+27x3+5x2+8x+4dx=4dx+1x+1dx+2x+2dx+3(x+2)2dx=4x+log|x+1|+2log|x+2|3x+2+C

The method of partial fractions is not just confined to the problem of integrating rational functions. There are other integrals — such as secxdx and sec3xdx — that can be transformed (via substitutions) into integrals of rational functions. We encountered both of these integrals in Sections 1.8 and 1.9 on trigonometric integrals and substitutions.

Example 1.10.5 secxdx

Solution

In this example, we integrate secx. It is not yet clear what this integral has to do with partial fractions. To get to a partial fractions computation, we first make one of our old substitutions.

secxdx=1cosxdxmassage the expression a little=cosxcos2xdxsubstitute u=sinxdu=cosxdx=duu21and use cos2x=1sin2x=1u2

So we now have to integrate 1u21, which is a rational function of u, and so is perfect for partial fractions.

  • Step 1. The degree of the numerator, 1, is zero, which is strictly smaller than the degree of the denominator, u21, which is two. So the first step is skipped.
  • Step 2. The second step is to factor the denominator:

    u21=(u1)(u+1)

  • Step 3. The third step is to write 1u21 in the form

    1u21=1(u1)(u+1)=Au1+Bu+1

    for some constants A and B.
  • Step 3 — Sneaky Method.
    • Multiply through by the denominator to get

      1=A(u+1)+B(u1)

      This equation must be true for all u.
    • If we now set u=1 then we eliminate B from the equation leaving us with

      1=2A so A=12.

    • Similarly, if we set u=1 then we eliminate A, leaving

      1=2Bwhich implies B=12.

    We have now found that A=12,B=12, so

    1u21=12[1u11u+1].

  • It is always a good idea to check our work.

    12u1+12u+1=12(u+1)12(u1)(u1)(u+1)=1(u1)(u+1)

  • Step 4. The final step is to integrate.

    secxdx=duu21after substitution=12duu1+12duu+1partial fractions=12log|u1|+12log|u+1|+C=12log|sin(x)1|+12log|sin(x)+1|+Crearrange a little=12log|1+sinx1sinx|+C

    Notice that since 1sinx1, we are free to drop the absolute values in the last line if we wish.

Another example in the same spirit, though a touch harder. Again, we saw this problem in Section 1.8 and 1.9.

Example 1.10.6 sec3xdx

Solution

  • We'll start by converting it into the integral of a rational function using the substitution u=sinx, du=cosxdx.
    \begin{align*} \int \sec^3 x\, d{x} &=\int \frac{1}{\cos^3 x}\, d{x} & \text{massage this a little}\\ &=\int \frac{\cos x}{\cos^4 x}\, d{x} &\text{replace $\cos^2x=1-\sin^2x=1-u^2$}\\ &=\int \frac{\cos x\, d{x}}
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    { [1-u^2]^2} \end{align*}
  • We could now find the partial fraction decomposition of the integrand 1[1u2]2 by executing the usual four steps. But it is easier to use

    1u21=12[1u11u+1]

    which we worked out in Example 1.10.5 above.
  • Squaring this gives

    1[1u2]2=14[1u11u+1]2=14[1(u1)22(u1)(u+1)+1(u+1)2]=14[1(u1)21u1+1u+1+1(u+1)2]

    where we have again used 1u21=12[1u11u+1] in the last step.
  • It only remains to do the integrals and simplify.

    sec3xdx=14[1(u1)21u1+1u+1+1(u+1)2]du=14[1u1log|u1|+log|u+1|1u+1]+Cgroup carefully=14[1u1+1u+1]+14[log|u+1|log|u1|]+Csum carefully=142uu21+14log|u+1u1|+Cclean up=12u1u2+14log|u+1u1|+Cput u=sinx=12sinxcos2x+14log|sinx+1sinx1|+C

The form of partial fraction decompositions

In the examples above we used the partial fractions method to decompose rational functions into easily integrated pieces. Each of those examples was quite involved and we had to spend quite a bit of time factoring and doing long division. The key step in each of the computations was Step 3 — in that step we decomposed the rational function N(x)D(x) (or R(x)D(x)), for which the degree of the numerator is strictly smaller than the degree of the denominator, into a sum of particularly simple rational functions, like Axa. We did not, however, give a systematic description of those decompositions.

In this subsection we fill that gap by describing the general 12 form of partial fraction decompositions. The justification of these forms is not part of the course, but the interested reader is invited to read the next (optional) subsection where such justification is given. In the following it is assumed that

  • N(x) and D(x) are polynomials with the degree of N(x) strictly smaller than the degree of D(x).
  • K is a constant.
  • a1, a2, , aj are all different numbers.
  • m1, m2, , mj, and n1, n2, , nk are all strictly positive integers.
  • x2+b1x+c1, x2+b2x+c2,   , x2+bkx+ck are all different.

Simple linear factor case

If the denominator D(x)=K(xa1)(xa2)(xaj) is a product of j different linear factors, then

Equation 1.10.7
N(x)D(x)=A1xa1+A2xa2++Ajxaj

We can then integrate each term

Axadx=Alog|xa|+C.

General linear factor case

If the denominator D(x)=K(xa1)m1(xa2)m2(xaj)mj then

Claim 1.10.8
N(x)D(x)=A1,1xa1+A1,2(xa1)2++A1,m1(xa1)m1=+A2,1xa2+A2,2(xa2)2++A2,m2(xa2)m2+=+Aj,1xaj+Aj,2(xaj)2++Aj,mj(xaj)mj

Notice that we could rewrite each line as

A1xa+A2(xa)2++Am(xa)m=A1(xa)m1+A2(xa)m2++Am(xa)m=B1xm1+B2xm2++Bm(xa)m

which is a polynomial whose degree, m1, is strictly smaller than that of the denominator (xa)m. But the form of Equation 1.10.8 is preferable because it is easier to integrate.

Axadx=Alog|xa|+CA(xa)kdx=1k1A(xa)k1provided k>1.

Simple linear and quadratic factor case

If D(x)=K(xa1)(xaj)(x2+b1x+c1)(x2+bkx+ck) then

Claim 1.10.9
N(x)D(x)=A1xa1++Ajxaj+B1x+C1x2+b1x+c1++Bkx+Ckx2+bkx+ck

Note that the numerator of each term on the right hand side has degree one smaller than the degree of the denominator.

The quadratic terms Bx+Cx2+bx+c are integrated in a two-step process that is best illustrated with a simple example (see also Example 1.10.3 above).

Example 1.10.10 2x+7x2+4x+13dx

Solution

  • Start by completing the square in the denominator:

    x2+4x+13=(x+2)2+9and thus2x+7x2+4x+13=2x+7(x+2)2+32

  • Now set y=(x+2)/3,dy=13dx, or equivalently x=3y-2, \, d{x}=3\, d{y}\text{:}

    \begin{align*} \int \frac{2x+7}{x^2+4x+13}\, d{x} &= \int \frac{2x+7}{(x+2)^2+3^2}\, d{x}\\ &= \int \frac{6y-4+7}{3^2 y^2 + 3^2} \cdot 3 \, d{y}\\ &= \int \frac{6y+3}{3(y^2+1)}\, d{y}\\ &= \int \frac{2y+1}{y^2+1}\, d{y} \end{align*}

    Notice that we chose 3 in y=(x+2)/3 precisely to transform the denominator into the form y^2+1\text{.}
  • Now almost always the numerator will be a linear polynomial of y and we decompose as follows

    \begin{align*} \int \frac{2x+7}{x^2+4x+13}\, d{x} &= \int \frac{2y+1}{y^2+1}\, d{y}\\ &= \int \frac{2y}{y^2+1}\, d{y} + \int \frac{1}{y^2+1}\, d{y}\\ &= \log|y^2+1| + \arctan y + C\\ &= \log\left|\left(\frac{x+2}{3}\right)^2 +1 \right| + \arctan\left(\frac{x+2}{3} \right)+C \end{align*}

Optional — General linear and quadratic factor case

If D(x)=K(x-a_1)^{m_1}\cdots(x-a_j)^{m_j} (x^2+b_1x+c_1)^{n_1}\cdots(x^2+b_kx+c_k)^{n_k}

Claim 1.10.11
\begin{align*} \frac{N(x)}{D(x)} &=\frac{A_{1,1}}{x-a_1}+\frac{A_{1,2}}{(x-a_1)^2}+\cdots +\frac{A_{1,m_1}}{(x-a_1)^{m_1}}+\cdots\\ &\phantom{=}\!+\frac{A_{j,1}}{x-a_j}+\frac{A_{j,2}}{(x-a_j)^2}+\cdots +\frac{A_{j,m_j}}{(x-a_j)^{m_j}}\\ &\phantom{=}\!+\frac{B_{1,1}x+C_{1,1}}{x^2+b_1x+c_1} +\frac{B_{1,2}x+C_{1,2}}{(x^2+b_1x+c_1)^2}+\!\cdots\! +\frac{B_{1,n_1}x+C_{1,n_1}}{(x^2+b_1x+c_1)^{n_1}}\!+\!\cdots\\ &\phantom{=}\!+\frac{B_{k,1}x+C_{k,1}}{x^2+b_kx+c_k} +\frac{B_{k,2}x+C_{k,2}}{(x^2+b_kx+c_k)^2}+\!\cdots\! +\frac{B_{k,n_k}x+C_{1,n_k}}{(x^2+b_kx+c_k)^{n_k}} \end{align*}

We have already seen how to integrate the simple and general linear terms, and the simple quadratic terms. Integrating general quadratic terms is not so straightforward.

Example 1.10.12 \int \frac{\, d{x}}{(x^2+1)^n}

This example is not so easy, so it should definitely be considered optional.

Solution

In what follows write

\begin{align*} I_n &= \int \frac{\, d{x}}{(x^2+1)^n}. \end{align*}

  • When n=1 we know that

    \begin{align*} \int \frac{\, d{x}}{x^2+1} &= \arctan x +C \end{align*}

  • Now assume that n \gt 1\text{,} then

    \begin{align*} \int \frac{1}{(x^2+1)^n}\, d{x} &= \int \frac{(x^2+1-x^2)}{(x^2+1)^n}\, d{x} & \text{sneaky}\\ &= \int \frac{1}{(x^2+1)^{n-1}}\, d{x} - \int \frac{x^2}{(x^2+1)^n}\, d{x}\\ &= I_{n-1} - \int \frac{x^2}{(x^2+1)^n}\, d{x} \end{align*}

    So we can write I_n in terms of I_{n-1} and this second integral.
  • We can use integration by parts to compute the second integral:

    \begin{align*} \int \frac{x^2}{(x^2+1)^n}\, d{x} &= \int \frac{x}{2} \cdot \frac{2x}{(x^2+1)^n}\, d{x} & \text{sneaky} \end{align*}

    We set u=x/2 and \, d{v}=\frac{2x}{(x^2+1)^n}\, d{x}\text{,} which gives \, d{u}=\frac{1}{2}\, d{x} and v=-\frac{1}{n-1} \cdot \frac{1}{(x^2+1)^{n-1}}\text{.} You can check v by differentiating. Integration by parts gives

    \begin{align*} & \int \frac{x}{2} \cdot \frac{2x}{(x^2+1)^n}\, d{x}\\ &\hskip0.5in= - \frac{x}{2(n-1)(x^2+1)^{n-1}} + \int \frac{\, d{x}}{2(n-1)(x^2+1)^{n-1}}\\ &\hskip0.5in= - \frac{x}{2(n-1)(x^2+1)^{n-1}} + \frac{1}{2(n-1)} \cdot I_{n-1} \end{align*}

  • Now put everything together:

    \begin{align*} I_n &= \int \frac{1}{(x^2+1)^n}\, d{x}\\ &= I_{n-1} + \frac{x}{2(n-1)(x^2+1)^{n-1}} - \frac{1}{2(n-1)} \cdot I_{n-1}\\ &= \frac{2n-3}{2(n-1)} I_{n-1} + \frac{x}{2(n-1)(x^2+1)^{n-1}} \end{align*}

  • We can then use this recurrence to write down I_n for the first few n\text{:}

    \begin{align*} I_2 &= \frac{1}{2} I_1 + \frac{x}{2 (x^2+1)}+C\\ &= \frac{1}{2}\arctan x + \frac{x}{2(x^2+1)}\\ I_3 &= \frac{3}{4} I_2 + \frac{x}{4(x^2+1)^2}\\ &= \frac{3}{8}\arctan x + \frac{3x}{8(x^2+1)} + \frac{x}{4(x^2+1)^2}+C\\ I_4 &= \frac{5}{6} I_3 + \frac{x}{6(x^2+1)^3}\\ &= \frac{5}{16}\arctan x + \frac{5x}{16(x^2+1)} + \frac{5x}{24(x^2+1)^2} + \frac{x}{6(x^2+1)^3}+C \end{align*}

    and so forth. You can see why partial fraction questions involving denominators with repeated quadratic factors do not often appear on exams.

Optional — Justification of the partial fraction decompositions

We will now see the justification for the form of the partial fraction decompositions. We will only consider the case in which the denominator has only linear factors. The arguments when there are quadratic factors too are similar 13.

Simple linear factor case

In the most common partial fraction decomposition, we split up

\begin{gather*} \frac{N(x)}{(x-a_1)\times\cdots\times (x-a_d)} \end{gather*}

into a sum of the form

\begin{gather*} \frac{A_1}{x-a_1}+\cdots+\frac{A_d}{x-a_d} \end{gather*}

We now show that this decomposition can always be achieved, under the assumptions that the a_i's are all different and N(x) is a polynomial of degree at most d-1\text{.} To do so, we shall repeatedly apply the following Lemma.

Lemma 1.10.13

Let N(x) and D(x) be polynomials of degree n and d respectively, with n\le d\text{.} Suppose that a is NOT a zero of D(x)\text{.} Then there is a polynomial P(x) of degree p \lt d and a number A such that

\begin{align*} \frac{N(x)}{D(x)\,(x-a)} &=\frac{P(x)}{D(x)}+\frac{A}{x-a} \end{align*}

Proof
  • To save writing, let z=x-a\text{.} We then write \tilde N(z)=N(z+a) and \tilde D(z)=D(z+a)\text{,} which are again polynomials of degree n and d respectively. We also know that \tilde D(0)=D(a)\ne 0\text{.}
  • In order to complete the proof we need to find a polynomial \tilde P(z) of degree p \lt d and a number A such that

    \begin{gather*} \frac{\tilde N(z)}{\tilde D(z)\,z} =\frac{\tilde P(z)}{\tilde D(z)}+\frac{A}{z} =\frac{\tilde P(z) z+A\tilde D(z)}{\tilde D(z)\,z} \end{gather*}

    or equivalently, such that

    \begin{gather*} \tilde P(z) z+A\tilde D(z)=\tilde N(z). \end{gather*}

  • Now look at the polynomial on the left hand side. Every term in \tilde P(z) z\text{,} has at least one power of z\text{.} So the constant term on the left hand side is exactly the constant term in A\tilde D(z)\text{,} which is equal to A\tilde D(0)\text{.} The constant term on the right hand side is equal to \tilde N(0)\text{.} So the constant terms on the left and right hand sides are the same if we choose A=\frac{\tilde N(0)}{\tilde D(0)}\text{.} Recall that \tilde D(0) cannot be zero, so A is well defined.
  • Now move A\tilde D(z) to the right hand side.

    \begin{gather*} \tilde P(z) z=\tilde N(z)-A\tilde D(z) \end{gather*}

    The constant terms in \tilde N(z) and A\tilde D(z) are the same, so the right hand side contains no constant term and the right hand side is of the form \tilde N_1(z) z for some polynomial \tilde N_1(z)\text{.}
  • Since \tilde N(z) is of degree at most d and A\tilde D(z) is of degree exactly d\text{,} \tilde N_1 is a polynomial of degree d-1\text{.} It now suffices to choose \tilde P(z)=\tilde N_1(z)\text{.}

Now back to

\begin{gather*} \frac{N(x)}{(x-a_1)\times\cdots\times (x-a_d)} \end{gather*}

Apply Lemma 1.10.13, with D(x)=(x-a_2)\times\cdots\times (x-a_d) and a=a_1\text{.} It says

\begin{gather*} \frac{N(x)}{(x-a_1)\times\cdots\times (x-a_d)} =\frac{A_1}{x-a_1}+\frac{P(x)}{(x-a_2)\times\cdots\times (x-a_d)} \end{gather*}

for some polynomial P of degree at most d-2 and some number A_1\text{.}

Apply Lemma 1.10.13 a second time, with D(x)=(x-a_3)\times\cdots\times (x-a_d)\text{,} N(x)=P(x) and a=a_2\text{.} It says

\begin{gather*} \frac{P(x)}{(x-a_2)\times\cdots\times (x-a_d)} =\frac{A_2}{x-a_2}+\frac{Q(x)}{(x-a_3)\times\cdots\times (x-a_d)} \end{gather*}

for some polynomial Q of degree at most d-3 and some number A_2\text{.}

At this stage, we know that

\begin{gather*} \frac{N(x)}{(x-a_1)\times\cdots\times (x-a_d)} =\frac{A_1}{x-a_1}+\frac{A_2}{x-a_2}+ \frac{Q(x)}{(x-a_3)\times\cdots\times (x-a_d)} \end{gather*}

If we just keep going, repeatedly applying Lemma 1, we eventually end up with

\begin{gather*} \frac{N(x)}{(x-a_1)\times\cdots\times (x-a_d)} =\frac{A_1}{x-a_1}+\cdots+\frac{A_d}{x-a_d} \end{gather*}

as required.

The general case with linear factors

Now consider splitting

\begin{gather*} \frac{N(x)}{(x-a_1)^{n_1}\times\cdots\times (x-a_d)^{n_d}} \end{gather*}

into a sum of the form 14

\begin{gather*} \Big[\frac{A_{1,1}}{x-a_1}+\cdots+\frac{A_{1,n_1}}{(x-a_1)^{n_1}}\Big]+\cdots+ \Big[\frac{A_{d,1}}{x-a_d}+\cdots+\frac{A_{d,n_d}}{(x-a_d)^{n_d}}\Big] \end{gather*}

We now show that this decomposition can always be achieved, under the assumptions that the a_i's are all different and N(x) is a polynomial of degree at most n_1+\cdots+n_d-1\text{.} To do so, we shall repeatedly apply the following Lemma.

Lemma 1.10.14

Let N(x) and D(x) be polynomials of degree n and d respectively, with n \lt d+m\text{.} Suppose that a is NOT a zero of D(x)\text{.} Then there is a polynomial P(x) of degree p \lt d and numbers A_1,\ \cdots,\ A_m such that

\begin{gather*} \frac{N(x)}{D(x)\,(x-a)^m} =\frac{P(x)}{D(x)}+\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots +\frac{A_m}{(x-a)^m} \end{gather*}

Proof
  • As we did in the proof of the previous lemma, we write z=x-a\text{.} Then \tilde N(z)=N(z+a) and \tilde D(z)=D(z+a) are polynomials of degree n and d respectively, \tilde D(0)=D(a)\ne 0\text{.}
  • In order to complete the proof we have to find a polynomial \tilde P(z) of degree p \lt d and numbers A_1,\ \cdots,\ A_m such that

    \begin{align*} \frac{\tilde N(z)}{\tilde D(z)\,z^m} &=\frac{\tilde P(z)}{\tilde D(z)}+\frac{A_1}{z}+\frac{A_2}{z^2}+\cdots +\frac{A_m}{z^m}\\ &=\frac{\tilde P(z) z^m+A_1z^{m-1}\tilde D(z)+A_2z^{m-2}\tilde D(z)+\cdots+A_m\tilde D(z)}{\tilde D(z)\,z^m} \end{align*}

    or equivalently, such that

    \begin{align*} &\tilde P(z) z^m+A_1z^{m-1}\tilde D(z)+A_2z^{m-2}\tilde D(z)+\cdots +A_{m-1}z\tilde D(z)+A_m\tilde D(z)\\ &\hskip3in=\tilde N(z) \end{align*}

  • Now look at the polynomial on the left hand side. Every single term on the left hand side, except for the very last one, A_m\tilde D(z)\text{,} has at least one power of z\text{.} So the constant term on the left hand side is exactly the constant term in A_m\tilde D(z)\text{,} which is equal to A_m\tilde D(0)\text{.} The constant term on the right hand side is equal to \tilde N(0)\text{.} So the constant terms on the left and right hand sides are the same if we choose A_m=\frac{\tilde N(0)}{\tilde D(0)}\text{.} Recall that \tilde D(0)\ne 0 so A_m is well defined.
  • Now move A_m\tilde D(z) to the right hand side.

    \begin{align*} &\tilde P(z) z^m+A_1z^{m-1}\tilde D(z)+A_2z^{m-2}\tilde D(z)+\cdots +A_{m-1}z\tilde D(z)\\ &\hskip3in=\tilde N(z)-A_m\tilde D(z) \end{align*}

    The constant terms in \tilde N(z) and A_m\tilde D(z) are the same, so the right hand side contains no constant term and the right hand side is of the form \tilde N_1(z) z with \tilde N_1 a polynomial of degree at most d+m-2\text{.} (Recall that \tilde N is of degree at most d+m-1 and \tilde D is of degree at most d\text{.}) Divide the whole equation by z to get

    \begin{gather*} \tilde P(z) z^{m-1}+A_1z^{m-2}\tilde D(z)+A_2z^{m-3}\tilde D(z)+\cdots +A_{m-1}\tilde D(z) =\tilde N_1(z). \end{gather*}

  • Now, we can repeat the previous argument. The constant term on the left hand side, which is exactly equal to A_{m-1}\tilde D(0) matches the constant term on the right hand side, which is equal to \tilde N_1(0) if we choose A_{m-1}=\frac{\tilde N_1(0)}{\tilde D(0)}\text{.} With this choice of A_{m-1}

    \begin{gather*} \tilde P(z) z^{m-1}+A_1z^{m-2}\tilde D(z)+A_2z^{m-3}\tilde D(z)+\cdots +A_{m-2} z\tilde D(z)\\ =\tilde N_1(z)-A_{m-1}\tilde D(z)=\tilde N_2(z)z \end{gather*}

    with \tilde N_2 a polynomial of degree at most d+m-3\text{.} Divide by z and continue.
  • After m steps like this, we end up with

    \begin{gather*} \tilde P(z) z=\tilde N_{m-1}(z)-A_1\tilde D(z) \end{gather*}

    after having chosen A_1=\frac{\tilde N_{m-1}(0)}{\tilde D(0)}\text{.}
  • There is no constant term on the right side so that \tilde N_{m-1}(z)-A_1\tilde D(z) is of the form \tilde N_m(z) z with \tilde N_m a polynomial of degree d-1\text{.} Choosing \tilde P(z)=\tilde N_m(z) completes the proof.

Now back to

\begin{gather*} \frac{N(x)}{(x-a_1)^{n_1}\times\cdots\times (x-a_d)^{n_d}} \end{gather*}

Apply Lemma 1.10.14, with D(x)=(x-a_2)^{n_2}\times\cdots\times (x-a_d)^{n_d}\text{,} m=n_1 and a=a_1\text{.} It says

\begin{align*} &\frac{N(x)}{(x-a_1)^{n_1}\times\cdots\times (x-a_d)^{n_d}}\\ & =\frac{A_{1,1}}{x-a_1}+\frac{A_{1,2}}{(x-a_1)^2}+\cdots +\frac{A_{1,n_1}}{(x-a)^{n_1}} +\frac{P(x)}{(x-a_2)^{n_2}\times\cdots\times (x-a_d)^{n_d}} \end{align*}

Apply Lemma 1.10.14 a second time, with D(x)=(x-a_3)^{n_3}\times\cdots\times (x-a_d)^{n_d}\text{,} N(x)=P(x)\text{,} m=n_2 and a=a_2\text{.} And so on. Eventually, we end up with

\begin{gather*} \Big[\frac{A_{1,1}}{x-a_1}+\cdots+\frac{A_{1,n_1}}{(x-a_1)^{n_1}}\Big]+\cdots+ \Big[\frac{A_{d,1}}{x-a_d}+\cdots+\frac{A_{d,n_d}}{(x-a_d)^{n_d}}\Big] \end{gather*}

which is exactly what we were trying to show.

Really Optional — The Fully General Case

We are now going to see that, in general, if N(x) and D(x) are polynomials with the degree of N being strictly smaller than the degree of D (which we'll denote \deg(N) \lt \deg(D)) and if

\begin{gather} D(x)=K(x-a_1)^{m_1}\cdots(x-a_j)^{m_j} (x^2+b_1x+c_1)^{n_1}\cdots(x^2+b_kx+c_k)^{n_k}\label{eq_INTfactD}\tag{\(\star\)} \end{gather}

(with b_\ell^2-4 c_\ell \lt 0 for all 1\le\ell\le k so that no quadratic factor can be written as a product of linear factors with real coefficients) then there are real numbers A_{i,j}\text{,} B_{i,j}\text{,} C_{i,j} such that

\begin{align*} \frac{N(x)}{D(x)} &=\frac{A_{1,1}}{x-a_1}+\frac{A_{1,2}}{(x-a_1)^2}+\cdots +\frac{A_{1,m_1}}{(x-a_1)^{m_1}}+\cdots\\ &\phantom{=}\!+\frac{A_{j,1}}{x-a_j}+\frac{A_{j,2}}{(x-a_j)^2}+\cdots +\frac{A_{j,m_j}}{(x-a_j)^{m_j}}\\ &\phantom{=}\!+\frac{B_{1,1}x+C_{1,1}}{x^2+b_1x+c_1} +\frac{B_{1,2}x+C_{1,2}}{(x^2+b_1x+c_1)^2}+\!\cdots\! +\frac{B_{1,n_1}x+C_{1,n_1}}{(x^2+b_1x+c_1)^{n_1}}\!+\!\cdots\\ &\phantom{=}\!+\frac{B_{k,1}x+C_{k,1}}{x^2+b_kx+c_k} +\frac{B_{k,2}x+C_{k,2}}{(x^2+b_kx+c_k)^2}+\!\cdots\! +\frac{B_{k,n_k}x+C_{1,n_k}}{(x^2+b_kx+c_k)^{n_k}} \end{align*}

This was Equation 1.10.11. We start with two simpler results, that we'll use repeatedly to get Equation 1.10.11. In the first simpler result, we consider the fraction \frac{P(x)}{Q_1(x)\,Q_2(x)} with P(x)\text{,} Q_1(x) and Q_2(x) being polynomials with real coefficients and we are going to assume that when P(x)\text{,} Q_1(x) and Q_2(x) are factored as in (\star), no two of them have a common linear or quadratic factor. As an example, no two of

\begin{align*} P(x) &= 2(x-3)(x-4)(x^2+3x+3)\\ Q_1(x) &= 2(x-1)(x^2+2x+2)\\ Q_2(x) &= 2(x-2)(x^2+2x+3) \end{align*}

have such a common factor. But, for

\begin{align*} P(x) &= 2(x-3)(x-4)(x^2+x+1)\\ Q_1(x) &= 2(x-1)(x^2+2x+2)\\ Q_2(x) &= 2(x-2)(x^2+x+1) \end{align*}

P(x) and Q_2(x) have the common factor x^2+x+1\text{.}

Lemma 1.10.15

Let P(x)\text{,} Q_1(x) and Q_2(x) be polynomials with real coefficients and with \deg(P)\lt\deg(Q_1Q_2)\text{.} Assume that no two of P(x)\text{,} Q_1(x) and Q_2(x) have a common linear or quadratic factor. Then there are polynomials P_1,\ P_2 with \deg(P_1)\lt\deg(Q_1)\text{,} \deg(P_2)\lt\deg(Q_2)\text{,} and

\begin{gather*} \frac{P(x)}{Q_1(x)\,Q_2(x)} =\frac{P_1(x)}{Q_1(x)} + \frac{P_2(x)}{Q_2(x)} \end{gather*}

Proof

We are to find polynomials P_1 and P_2 that obey

\begin{gather*} P(x) = P_1(x)\,Q_2(x) + P_2(x)\,Q_1(x) \end{gather*}

Actually, we are going to find polynomials p_1 and p_2 that obey

\begin{gather} p_1(x)\,Q_1(x) +p_2(x)\,Q_2(x) = C\label{eq_INTponeptwo}\tag{\(\star\star\)} \end{gather}

for some nonzero constant C\text{,} and then just multiply (\star\star) by \frac{P(x)}{C}\text{.} To find p_1\text{,} p_2 and C we are going to use something called the Euclidean algorithm. It is an algorithm 15 that is used to efficiently find the greatest common divisors of two numbers. Because Q_1(x) and Q_2(x) have no common factors of degree 1 or 2\text{,} their "greatest common divisor'' has degree 0\text{,} i.e. is a constant.

  • The first step is to apply long division to \frac{Q_1(x)}{Q_2(x)} to find polynomials n_0(x) and r_0(x) such that

    \begin{gather*} \frac{Q_1(x)}{Q_2(x)} = n_0(x) +\frac{r_0(x)}{Q_2(x)}\qquad \text{with }\deg(r_0)\lt\deg(Q_2) \end{gather*}

    or, equivalently,

    \begin{gather*} Q_1(x) = n_0(x)\,Q_2(x) + r_0(x)\qquad \text{with }\deg(r_0)\lt\deg(Q_2) \end{gather*}

  • The second step is to apply long division to \frac{Q_2(x)}{r_0(x)} to find polynomials n_1(x) and r_1(x) such that

    \begin{gather*} Q_2(x) = n_1(x)\,r_0(x) + r_1(x)\qquad \text{with }\deg(r_1)\lt\deg(r_0)\ \ \text{or}\ \ r_1(x)=0 \end{gather*}

  • The third step (assuming that r_1(x) was not zero) is to apply long division to \frac{r_0(x)}{r_1(x)} to find polynomials n_2(x) and r_2(x) such that

    \begin{gather*} r_0(x) = n_2(x)\,r_1(x) + r_2(x)\qquad \text{with }\deg(r_2)\lt\deg(r_1)\ \ \text{or}\ \ r_2(x)=0 \end{gather*}

  • And so on.

As the degree of the remainder r_i(x) decreases by at least one each time i is increased by one, the above iteration has to terminate with some r_{\ell+1}(x)=0\text{.} That is, we choose \ell to be index of the last nonzero remainder. Here is a summary of all of the long division steps.

\begin{align*} Q_1(x)&=n_0(x)\, Q_2(x) + r_0(x)\qquad &&\text{with }\deg(r_0)\lt\deg(Q_2)\\ Q_2(x)&=n_1(x)\, r_0(x) + r_1(x)\qquad &&\text{with }\deg(r_1)\lt\deg(r_0)\\ r_0(x)&=n_2(x)\,r_1(x)+r_2(x)\qquad &&\text{with }\deg(r_2)\lt\deg(r_1)\\ r_1(x)&=n_3(x)\,r_2(x)+r_3(x)\qquad &&\text{with }\deg(r_3)\lt\deg(r_2)\\ &\ \ \vdots\\ r_{\ell-2}(x)&=n_\ell(x)\,r_{\ell-1}(x)+r_\ell(x)\qquad &&\text{with }\deg(r_\ell)\lt\deg(r_{\ell-1})\\ r_{\ell-1}(x)&=n_{\ell+1}(x)\,r_\ell(x)+r_{\ell+1}(x)\qquad &&\text{with }r_{\ell+1}=0 \end{align*}

Now we are going to take a closer look at all of the different remainders that we have generated.

  • From first long division step, namely Q_1(x) = n_0(x)\,Q_2(x) + r_0(x) we have that the remainder

    \begin{gather*} r_0(x)=Q_1(x)-n_0(x)\,Q_2(x) \end{gather*}

  • From the second long division step, namely Q_2(x) = n_1(x)\,r_0(x) + r_1(x) we have that the remainder

    \begin{align*} & r_1(x)=Q_2(x)-n_1(x)\,r_0(x) = Q_2(x)-n_1(x)\big[Q_1(x)-n_0(x)\,Q_2(x)\big]\\ &\phantom{r_1(x)} =A_1(x)\,Q_1(x)+B_1(x)\,Q_2(x) \end{align*}

    with A_1(x) =-n_1(x) and B_1(x) = 1+n_0(x)\,n_1(x)\text{.}
  • From the third long division step (assuming that r_1(x) was not zero), namely r_0(x)=n_2(x)\,r_1(x)+r_2(x)\text{,} we have that the remainder

    \begin{align*} r_2(x)&=r_0(x)-n_2(x)\,r_1(x)\\ &=\big[Q_1(x)-n_0(x)\,Q_2(x)\big] -n_2(x)\big[A_1(x)\,Q_1(x)+B_1(x)\,Q_2(x)\big]\\ &=A_2(x)\,Q_1(x)+B_2(x)\,Q_2(x) \end{align*}

    with A_2(x)= 1-n_2(x)\,A_1(x) and B_2(x) = -n_0(x)-n_2(x)\,B_1(x)\text{.}
  • And so on. Continuing in this way, we conclude that the final nonzero remainder r_\ell(x)=A_\ell(x)\,Q_1(x)+B_\ell(x)\,Q_2(x) for some polynomials A_\ell and B_\ell\text{.}

Now the last nonzero remainder r_\ell(x) has to be a nonzero constant C because

  • it is nonzero by the definition of r_\ell(x) and
  • if r_\ell(x) were a polynomial of degree at least one, then
  • so that r_\ell(x) would be a common factor for Q_1(x) and Q_2(x)\text{,} in contradiction to the hypothesis that no two of P(x)\text{,} Q_1(x) and Q_2(x) have a common linear or quadratic factor.

We now have that A_\ell(x)\,Q_1(x)+B_\ell(x)\,Q_2(x)=r_\ell(x)=C\text{.} Multiplying by \frac{P(x)}{C} gives

\begin{gather*} \tilde P_2(x)\,Q_1(x)+\tilde P_1(x)\,Q_2(x)=P(x)\quad\text{or}\quad \frac{\tilde P_1(x)}{Q_1(x)} +\frac{\tilde P_2(x)}{Q_2(x)} =\frac{P(x)}{Q_1(x)\,Q_2(x)} \end{gather*}

with \tilde P_2(x)=\frac{P(x)\,A_\ell(x)}{C} and \tilde P_1(x)=\frac{P(x)\,B_\ell(x)}{C}\text{.} We're not quite done, because there is still the danger that \deg(\tilde P_1)\ge \deg(Q_1) or \deg(\tilde P_2)\ge \deg(Q_2)\text{.} To deal with that possibility, we long divide \frac{\tilde P_1(x)}{Q_1(x)} and call the remainder P_1(x)\text{.}

\begin{gather*} \frac{\tilde P_1(x)}{Q_1(x)}=N(x)+\frac{P_1(x)}{Q_1(x)} \qquad\text{ with }\deg(P_1)\lt\deg(Q_1) \end{gather*}

Therefore we have that

\begin{align*} \frac{P(x)}{Q_1(x)\,Q_2(x)} &=\frac{P_1(x)}{Q_1(x)} +N(x) +\frac{\tilde P_2(x)}{Q_2(x)}\\ &=\frac{P_1(x)}{Q_1(x)} +\frac{\tilde P_2(x)+N(x)Q_2(x)}{Q_2(x)} \end{align*}

Denoting P_2(x)=\tilde P_2(x)+N(x)Q_2(x) gives \frac{P}{Q_1\,Q_2} =\frac{P_1}{Q_1} +\frac{P_2}{Q_2} and since \deg(P_1)\lt\deg(Q_1)\text{,} the only thing left to prove is that \deg(P_2)\lt\deg(Q_2)\text{.} We assume that \deg(P_2)\ge\deg(Q_2) and look for a contradiction. We have

\begin{align*} &\deg(P_2Q_1)\ge\deg(Q_1Q_2)\gt\deg(P_1Q_2)\\ &\implies \deg(P)=\deg(P_1Q_2+P_2Q_1)=\deg(P_2Q_1)\ge\deg(Q_1Q_2) \end{align*}

which contradicts the hypothesis that \deg(P)\lt\deg(Q_1Q_2) and the proof is complete.

For the second of the two simpler results, that we'll shortly use repeatedly to get Equation 1.10.11, we consider \frac{P(x)}{(x-a)^m} and \frac{P(x)}{{(x^2+bx+c)}^m}\text{.}

Lemma 1.10.16

Let m\ge 2 be an integer, and let Q(x) be either x-a or x^2+bx+c\text{,} with a\text{,} b and c being real numbers. Let P(x) be a polynomial with real coefficients, which does not contain Q(x) as a factor, and with \deg(P)\lt\deg(Q^m)=m\deg(Q)\text{.} Then, for each 1\le i\le m\text{,} there is a polynomial P_i with \deg(P_i)\lt\deg(Q) or P_i=0\text{,} such that

\begin{gather*} \frac{P(x)}{Q(x)^m} =\frac{P_1(x)}{Q(x)}+\frac{P_2(x)}{Q(x)^2}+\frac{P_3(x)}{Q(x)^3}+\cdots +\frac{P_{m-1}(x)}{Q(x)^{m-1}}+\frac{P_{m}(x)}{Q(x)^m}. \end{gather*}

In particular, if Q(x) =x-a\text{,} then each P_i(x) is just a constant A_i\text{,} and if Q(x) =x^2+bx+c\text{,} then each P_i(x) is a polynomial B_i x+ C_i of degree at most one.

Proof

We simply repeatedly use long divison to get

\begin{align*} \frac{P(x)}{Q(x)^m} &=\frac{P(x)}{Q(x)}\,\frac{1}{Q(x)^{m-1}} =\left\{n_1(x) + \frac{r_1(x)}{Q(x)}\right\}\frac{1}{Q(x)^{m-1}}\\ &= \frac{r_1(x)}{Q(x)^m}+\frac{n_1(x)}{Q(x)}\,\frac{1}{Q(x)^{m-2}}\\ &= \frac{r_1(x)}{Q(x)^m} +\left\{n_2(x) + \frac{r_2(x)}{Q(x)}\right\}\frac{1}{Q(x)^{m-2}}\\ &= \frac{r_1(x)}{Q(x)^m} + \frac{r_2(x)}{Q(x)^{m-1}}+\frac{n_2(x)}{Q(x)}\,\frac{1}{Q(x)^{m-3}}\\ &\ \ \vdots\\ &= \frac{r_1(x)}{Q(x)^m} + \frac{r_2(x)}{Q(x)^{m-1}} +\cdots+ \frac{r_{m-2}(x)}{Q(x)^3} +\frac{n_{m-2}(x)}{Q(x)}\,\frac{1}{Q(x)}\\ &= \frac{r_1(x)}{Q(x)^m} + \frac{r_2(x)}{Q(x)^{m-1}} +\cdots+ \frac{r_{m-2}(x)}{Q(x)^3} +\\ &\hskip1in\left\{n_{m-1}(x) + \frac{r_{m-1}(x)}{Q(x)}\right\}\frac{1}{Q(x)}\\ &= \frac{r_1(x)}{Q(x)^m} + \frac{r_2(x)}{Q(x)^{m-1}} +\cdots+ \frac{r_{m-2}(x)}{Q(x)^3}+\frac{r_{m-1}(x)}{Q(x)^2} +\frac{n_{m-1}(x)}{Q(x)} \end{align*}

By the rules of long division every \deg(r_i)\lt\deg(Q)\text{.} It is also true that the final numerator, n_{m-1}\text{,} has \deg(n_{m-1})\lt\deg(Q) — that is, we kept dividing by Q until the degree of the quotient was less than the degree of Q\text{.} To see this, note that \deg(P)\lt m\deg(Q) and

\begin{align*} \deg(n_1) &= \deg(P)-\deg(Q)\\ \deg(n_2) &= \deg(n_1)-\deg(Q)= \deg(P)-2\deg(Q)\\ &\ \ \vdots\\ \deg(n_{m-1}) &= \deg(n_{m-2})-\deg(Q)= \deg(P)-(m-1)\deg(Q) \\ &\lt m\deg(Q)-(m-1)\deg(Q) \\ &=\deg(Q) \end{align*}

So, if \deg(Q)=1\text{,} then r_1, r_2, \ldots, r_{m-1}, n_{m-1} are all real numbers, and if \deg(Q)=2\text{,} then r_1, r_2, \ldots, r_{m-1}, n_{m-1} all have degree at most one.

We are now in a position to get Equation 1.10.11. We use (\star) to factor 16 D(x)= (x-a_1)^{m_1} Q_2(x) and use Lemma 1.10.15 to get

\begin{gather*} \frac{N(x)}{D(x)} =\frac{N(x)}{(x-a_1)^{m_1} Q_2(x)}=\frac{P_1(x)}{(x-a_1)^{m_1}} + \frac{P_2(x)}{Q_2(x)} \end{gather*}

where \deg(P_1)\lt m_1\text{,} and \deg(P_2)\lt\deg(Q_2)\text{.} Then we use Lemma Lemma 1.10.16 to get

\begin{gather*} \frac{N(x)}{D(x)} =\frac{P_1(x)}{(x-a_1)^{m_1}} + \frac{P_2(x)}{Q_2(x)} =\frac{A_{1,1}}{x-a_1}+\frac{A_{1,2}}{(x-a_1)^2}+\cdots +\frac{A_{1,m_1}}{(x-a_1)^{m_1}} + \frac{P_2(x)}{Q_2(x)} \end{gather*}

We continue working on \frac{P_2(x)}{Q_2(x)} in this way, pulling off of the denominator one (x-a_i)^{m_i} or one (x^2+b_ix + c_i)^{n_i} at a time, until we exhaust all of the factors in the denominator D(x)\text{.}

Exercises

Recall that we are using \log x to denote the logarithm of x with base e\text{.} In other courses it is often denoted \ln x\text{.}

Stage 1
1

Below are the graphs of four different quadratic functions. For each quadratic function, decide whether it is: (i) irreducible, (ii) the product of two distinct linear factors, or (iii) the product of a repeated linear factor (and possibly a constant).

image-263.svg        image-264.svg        image-265.svg        image-266.svg

2 (✳)

Write out the general form of the partial-fractions decomposition of \displaystyle\frac{x^3+3}{(x^2-1)^2(x^2+1)} \text{.} You need not determine the values of any of the coefficients.

3 (✳)

Find the coefficient of \displaystyle \frac{1}{x-1} in the partial fraction decomposition of \displaystyle\frac{3x^3-2x^2+11}{x^2(x-1)(x^2+3)}\text{.}

4

Re-write the following rational functions as the sum of a polynomial and a rational function whose numerator has a strictly smaller degree than its denominator. (Remember our method of partial fraction decomposition of a rational function only works when the degree of the numerator is strictly smaller than the degree of the denominator.)

  1. \dfrac{x^3+2x+2}{x^2+1}
  2. \dfrac{15x^4+6x^3+34x^2+4x+20}{5x^2+2x+8}
  3. \dfrac{2x^5+9x^3+12x^2+10x+30}{2x^2+5}
5

Factor the following polynomials into linear and irreducible factors.

  1. 5x^3-3x^2-10x+6
  2. x^4-3x^2-5
  3. x^4-4x^3-10x^2-11x-6
  4. 2x^4+12x^3-x^2-52x+15
6

Here is a fact:

Suppose we have a rational function with a repeated linear factor (ax+b)^n in the denominator, and the degree of the numerator is strictly less than the degree of the denominator. In the partial fraction decomposition, we can replace the terms

\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2}+\frac{A_3}{(ax+b)^3}+\cdots+ \frac{A_n}{(ax+b)^n}\tag{1} \nonumber

with the single term

\frac{B_1+B_2x+B_3x^2+\cdots + B_{n}x^{n-1}}{(ax+b)^n}\tag{2} \nonumber

and still be guaranteed to find a solution.

Why do we use the sum in (1), rather than the single term in (2), in partial fraction decomposition?

Stage 2
7 (✳)

Evaluate \displaystyle\int_1^2 \frac{\, d{x}}{x+x^2}\text{.}

8 (✳)

Calculate \displaystyle \int \frac{1}{x^4+x^2}\,\, d{x}\text{.}

9 (✳)

Calculate \displaystyle \int \frac{12x+4}{(x-3)(x^2+1)}\,dx\text{.}

10 (✳)

Evaluate the following indefinite integral using partial fraction:

\begin{gather*} F(x) = \int \frac{3x^2 -4}{(x-2)(x^2+4)}\,\, d{x} . \end{gather*}

11 (✳)

Evaluate \displaystyle\int \frac{x-13}{x^2-x-6}\, d{x}\text{.}

12 (✳)

Evaluate \displaystyle\int \frac{5x+1}{x^2+5x+6}\, d{x}\text{.}

13

Evaluate \displaystyle\int \frac{5x^2-3x-1}{x^2-1} \, d{x}\text{.}

14

Evaluate \displaystyle\int \frac{4x^4+14x^2+2}{4x^4+x^2} \, d{x}\text{.}

15

Evaluate \displaystyle\int \frac{x^2+2x-1}{x^4-2x^3+x^2} \, d{x}\text{.}

16

Evaluate \displaystyle\int \frac{ 3x^2-4x-10}{2x^3-x^2-8x+4} \, d{x}\text{.}

17

Evaluate \displaystyle\int_0^1 \frac{10x^2+24x+8}{2x^3+11x^2+6x+5} \, d{x}\text{.}

Stage 3

In Questions 18 and 19, we use partial fraction to find the antiderivatives of two important functions: cosecant, and cosecant cubed.

The purpose of performing a partial fraction decomposition is to manipulate an integrand into a form that is easily integrable. These “easily integrable” forms are rational functions whose denominator is a power of a linear function, or of an irreducible quadratic function. In Questions 20 through 23, we explore the integration of rational functions whose denominators involve irreducible quadratics.

In Questions 24 through 26, we use substitution to turn a non-rational integrand into a rational integrand, then evaluate the resulting integral using partial fraction. Till now, the partial fraction problems you've seen have all looked largely the same, but keep in mind that a partial fraction decomposition can be a small step in a larger problem.

18

Using the method of Example 1.10.5, integrate \displaystyle\int \csc x \, d{x}\text{.}

19

Using the method of Example 1.10.6, integrate \displaystyle\int \csc^3 x \, d{x}\text{.}

20

Evaluate \displaystyle\int_1^2 \frac{3x^3+15x^2+35x+10}{x^4+5x^3+10x^2} \, d{x}\text{.}

21

Evaluate \displaystyle\int\left(\frac{3}{x^2+2}+\frac{x-3}{(x^2+2)^2}\right) \, d{x} \text{.}

22

Evaluate \displaystyle\int\frac{1}{(1+x^2)^3} \, d{x}\text{.}

23

Evaluate \displaystyle\int \left(3x+\frac{3x+1}{x^2+5}+\frac{3x}{(x^2+5)^2}\right) \, d{x}\text{.}

24

Evaluate \displaystyle\int \frac{\cos\theta}{3\sin\theta+\cos^2\theta-3} \, d{\theta}\text{.}

25

Evaluate \displaystyle\int\frac{1}{e^{2t}+e^t+1} \, d{t}\text{.}

26

Evaluate \displaystyle\int\sqrt{1+e^x} \, d{x} using partial fraction.

27 (✳)

The region R is the portion of the first quadrant where 3\le x\le 4 and 0\le y\le\dfrac{10}{\sqrt{25-x^2}}\text{.}

  1. Sketch the region R\text{.}
  2. Determine the volume of the solid obtained by revolving R around the x-axis.
  3. Determine the volume of the solid obtained by revolving R around the y-axis.
28

Find the area of the finite region bounded by the curves y=\dfrac{4}{3+x^2}\text{,} y=\dfrac{2}{x(x+1)}\text{,} x=\dfrac14\text{,} and x=3\text{.}

29

Let F(x) = \displaystyle\int_1^x \frac{1}{t^2-9} \, d{t}\text{.}

  1. Give a formula for F(x) that does not involve an integral.
  2. Find F'(x)\text{.}
  1. Recall that a rational function is the ratio of two polynomials.
  2. The degree of a polynomial is the largest power of x\text{.} For example, the degree of 2x^3+4x^2+6x+8 is three.
  3. We will soon get to an example (Example 1.10.2 in fact) in which the numerator degree is at least as large as the denominator degree — in that situation we have to extract a polynomial P(x) before we can move on to step 2.
  4. That is, we take the decomposed form and sum it back together.
  5. Though, in fairness, we did step 3 twice — and that is the most tedious bit… Actually — sometimes factoring the denominator can be quite challenging. We'll consider this issue in more detail shortly.
  6. One does not typically think of mathematics assignments or exams as nice kind places… The polynomials that appear in the “real world” are not so forgiving. Nature, red in tooth and claw — to quote Tennyson inappropriately (especially when this author doesn't know any other words from the poem).
  7. Appendix A.16 contains several simple tricks for factoring polynomials. We recommend that you have a look at them.
  8. To be precise, the quadratic equation ax^2+bx+c=0 has solutions x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}. The term b^2-4ac is called the discriminant and it tells us about the number of solutions. If the discriminant is positive then there are two real solutions. When it is zero, there is a single solution. And if it is negative, there is no real solutions (you need complex numbers to say more than this).
  9. This same idea arose in Section 1.9. Given a quadratic written as Q(x)= ax^2+bx+c rewrite it as Q(x) = a(x+d)^2+e\text{.} We can determine d and e by expanding and comparing coefficients of x\text{:} ax^2+bx+c = a(x^2+2dx+d^2)+e = ax^2 + 2dax + (e+ad^2) \text{.} Hence d=b/2a and e=c-ad^2\text{.}
  10. At the risk of quoting Nietzsche, “That which does not kill us makes us stronger.” Though this author always preferred the logically equivalent contrapositive — “That which does not make us stronger will kill us.” However no one is likely to be injured by practicing partial fractions or looking up quotes on Wikipedia. Its also a good excuse to remind yourself of what a contrapositive is — though we will likely look at them again when we get to sequences and series.
  11. This is justified in the (optional) subsection “Justification of the Partial Fraction Decompositions” below.
  12. Well — not the completely general form, in the sense that we are not allowing the use of complex numbers. As a result we have to use both linear and quadratic factors in the denominator. If we could use complex numbers we would be able to restrict ourselves to linear factors.
  13. In fact, quadratic factors are completely avoidable because, if we use complex numbers, then every polynomial can be written as a product of linear factors. This is the fundamental theorem of algebra.
  14. If we allow ourselves to use complex numbers as roots, this is the general case. We don't need to consider quadratic (or higher) factors since all polynomials can be written as products of linear factors with complex coefficients.
  15. It appears in Euclid's Elements, which was written about 300 BC, and it was probably known even before that.
  16. This is assuming that there is at least one linear factor. If not, we factor D(x) = (x^2+b_1x + c_1)^{n_1} Q_2(x) instead.

This page titled 1.10: Partial Fractions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform.

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