Processing math: 79%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

1.9: Trigonometric Substitution

( \newcommand{\kernel}{\mathrm{null}\,}\)

In this section we discuss substitutions that simplify integrals containing square roots of the form

a2x2a2+x2x2a2.

When the integrand contains one of these square roots, then we can use trigonometric substitutions to eliminate them. That is, we substitute

x=asinuorx=atanuorx=asecu

and then use trigonometric identities

sin2θ+cos2θ=1and1+tan2θ=sec2θ

to simplify the result. To be more precise, we can

  • eliminate a2x2 from an integrand by substituting x=asinu to give

    a2x2=a2a2sin2u=a2cos2u=|acosu|

  • eliminate a2+x2 from an integrand by substituting x=atanu to give

    a2+x2=a2+a2tan2u=a2sec2u=|asecu|

  • eliminate x2a2 from an integrand by substituting x=asecu to give

    x2a2=a2sec2ua2=a2tan2u=|atanu|

    Be very careful with signs and absolute values when using this substitution. See Example 1.9.6.

When we have used substitutions before, we usually gave the new integration variable, u, as a function of the old integration variable x. Here we are doing the reverse — we are giving the old integration variable, x, in terms of the new integration variable u. We may do so, as long as we may invert to get u as a function of x. For example, with x=asinu, we may take u=arcsinxa. This is a good time for you to review the definitions of arcsinθ, arctanθ and arcsecθ. See Section 2.12, “Inverse Functions”, of the CLP-1 text.

As a warm-up, consider the area of a quarter of the unit circle.

Example 1.9.1 Quarter of the unit circle

Compute the area of the unit circle lying in the first quadrant.

Solution

We know that the answer is π4, but we can also compute this as an integral — we saw this way back in Example 1.1.16:

area=101x2dx

  • To simplify the integrand we substitute x=sinu. With this choice dxdu=cosu and so dx=cosudu.
  • We also need to translate the limits of integration and it is perhaps easiest to do this by writing u as a function of x — namely u(x)=arcsinx. Hence u(0)=0 and u(1)=π2.
  • Hence the integral becomes

    101x2dx=π201sin2ucosudu=π20cos2ucosudu=π20cos2udu

    Notice that here we have used that the positive square root cos2u=|cosu|=cosu because cos(u)0 for 0uπ2.
  • To go further we use the techniques of Section 1.8.

    101x2dx=π20cos2uduand since cos2u=1+cos2u2=12π20(1+cos(2u))du=12[u+12sin(2u)]π20=12(π20+sinπ2sin02)=π4

Example 1.9.2 x21x2dx

Solution

We proceed much as we did in the previous example.

  • To simplify the integrand we substitute x=sinu. With this choice dxdu=cosu and so dx=cosudu. Also note that u=arcsinx.
  • The integral becomes

    x21x2dx=sin2u1sin2ucosudu=sin2ucos2ucosudu

  • To proceed further we need to get rid of the square-root. Since u=arcsinx has domain 1x1 and range π2uπ2, it follows that cosu0 (since cosine is non-negative on these inputs). Hence

    cos2u=cosuwhen π2uπ2

  • So our integral now becomes

    x21x2dx=sin2ucos2ucosudu=sin2ucosucosudu=sin2udu=12(1cos2u)duby Equation 1.8.4=u214sin2u+C=12arcsinx14sin(2arcsinx)+C

  • We can simplify this further using a double-angle identity. Recall that u=arcsinx and that x=sinu. Then sin2u=2sinucosu

    We can replace cosu using cos2u=1sin2u. Taking a square-root of this formula gives cosu=±1sin2u. We need the positive branch here since cosu0 when π2uπ2 (which is exactly the range of arcsinx). Continuing along:

    sin2u=2sinu1sin2u=2x1x2 Thus our solution is

    x21x2dx=12arcsinx14sin(2arcsinx)+C=12arcsinx12x1x2+C

The above two example illustrate the main steps of the approach. The next example is similar, but with more complicated limits of integration.

Example 1.9.3 rar2x2dx

Let's find the area of the shaded region in the sketch below.

We'll set up the integral using vertical strips. The strip in the figure has width dx and height r2x2. So the area is given by the integral

area=rar2x2dx

Which is very similar to the previous example.

Solution

  • To evaluate the integral we substitute

    x=x(u)=rsinudx=dxdudu=rcosudu

    It is also helpful to write u as a function of x — namely u=arcsinxr.
  • The integral runs from x=a to x=r. These correspond to

    u(r)=arcsinrr=arcsin1=π2u(a)=arcsinar which does not simplify further

  • The integral then becomes

    rar2x2dx=π2arcsin(a/r)r2r2sin2urcosudu=π2arcsin(a/r)r21sin2ucosudu=r2π2arcsin(a/r)cos2ucosudu

    To proceed further (as we did in Examples 1.9.1 and 1.9.2) we need to think about whether cosu is positive or negative.
  • Since a (as shown in the diagram) satisfies 0ar, we know that u(a) lies between arcsin(0)=0 and arcsin(1)=π2. Hence the variable u lies between 0 and π2, and on this range cosu0. This allows us get rid of the square-root:

    cos2u=|cosu|=cosu

  • Putting this fact into our integral we get rar2x2dx=r2π2arcsin(a/r)cos2ucosudu=r2π2arcsin(a/r)cos2udu

    Recall the identity cos2u=1+cos2u2 from Section 1.8

    =r22π2arcsin(a/r)(1+cos2u)du=r22[u+12sin(2u)]π2arcsin(a/r)=r22(π2+12sinπarcsin(a/r)12sin(2arcsin(a/r)))=r22(π2arcsin(a/r)12sin(2arcsin(a/r))) Oof! But there is a little further to go before we are done.
  • We can again simplify the term sin(2arcsin(a/r)) using a double angle identity. Set θ=arcsin(a/r). Then θ is the angle in the triangle on the right below. By the double angle formula for sin(2θ) (Equation 1.8.2)

    sin(2θ)=2sinθ cosθ=2 ar r2a2r.

  • So finally the area is

    area=rar2x2dx=r22(π2arcsin(a/r)12sin(2arcsin(a/r)))=πr24r22arcsin(a/r)a2r2a2

  • This is a relatively complicated formula, but we can make some “reasonableness” checks, by looking at special values of a.
    • If a=0 the shaded region, in the figure at the beginning of this example, is exactly one quarter of a disk of radius r and so has area 14πr2. Substituting a=0 into our answer does indeed give 14πr2.
    • At the other extreme, if a=r, the shaded region disappears completely and so has area 0. Subbing a=r into our answer does indeed give 0, since arcsin1=π2.
Example 1.9.4 raxr2x2dx

The integral raxr2x2dx looks a lot like the integral we just did in the previous 3 examples. It can also be evaluated using the trigonometric substitution x=rsinu — but that is unnecessarily complicated. Just because you have now learned how to use trigonometric substitution 1 doesn't mean that you should forget everything you learned before.

Solution

This integral is much more easily evaluated using the simple substitution u=r2x2.

  • Set u=r2x2. Then du=2xdx, and so

    raxr2x2dx=0r2a2u du2=12[u3/23/2]0r2a2=13[r2a2]3/2

Enough sines and cosines — let us try a tangent substitution.

Example 1.9.5 dxx29+x2

Solution

As per our guidelines at the start of this section, the presence of the square root term 32+x2 tells us to substitute x=3tanu.

  • Substitute

    x=3tanudx=3sec2udu

    This allows us to remove the square root:

    9+x2=9+9tan2u=31+tan2u=3sec2u=3|secu|

  • Hence our integral becomes

    dxx29+x2=3sec2u9tan2u3|secu|du

  • To remove the absolute value we must consider the range of values of u in the integral. Since x=3tanu we have u=arctan(x/3). The range 2 of arctangent is π2arctanπ2 and so u=arctan(x/3) will always like between π2 and +π2. Hence cosu will always be positive, which in turn implies that |secu|=secu.
  • Using this fact our integral becomes:

    dxx29+x2=3sec2u27tan2u|secu|du=19secutan2udusince secu>0

  • Rewrite this in terms of sine and cosine dxx29+x2=19secutan2udu=191cosucos2usin2udu=19cosusin2udu

    Now we can use the substitution rule with y=sinu and dy=cosudu

    =19dyy2=19y+C=19sinu+C
  • The original integral was a function of x, so we still have to rewrite sinu in terms of x. Remember that x=3tanu or u=arctan(x/3). So u is the angle shown in the triangle below and we can read off the triangle that

    sinu=x9+x2dxx29+x2=9+x29x+C

Example 1.9.6 x2x21dx

Solution

This one requires a secant substitution, but otherwise is very similar to those above.

  • Set x=secu and dx=secutanudu. Then

    x2x21dx=sec2usec2u1secutanudu=sec3utanutan2udusince tan2u=sec2u1=sec3utanu|tanu|du

  • As before we need to consider the range of u values in order to determine the sign of tanu. Notice that the integrand is only defined when either x<1 or x>1; thus we should treat the cases x<1 and x>1 separately. Let us assume that x>1 and we will come back to the case x<1 at the end of the example.

    When x>1, our u=arcsecx takes values in (0,π2). This follows since when 0<u<π2, we have 0<cosu<1 and so secu>1. Further, when 0<u<π2, we have tanu>0. Thus |tanu|=tanu.

  • Back to our integral, when x>1: x2x21dx=sec3utanu|tanu|du=sec3udusince tanu0

    This is exactly Example 1.8.22

    =12secutanu+12log|secu+tanu|+C
  • Since we started with a function of x we need to finish with one. We know that secu=x and then we can use trig identities tan2u=sec2u1=x21so tanu=±x21

    but we know

    tanu0so tanu=x21 Thus

    x2x21dx=12xx21+12log|x+x21|+C

  • The above holds when x>1. We can confirm that it is also true when x<1 by showing the right-hand side is a valid antiderivative of the integrand. To do so we must differentiate our answer. Notice that we do not need to consider the sign of x+x21 when we differentiate since we have already seen that

    ddxlog|x|=1x

    when either x<0 or x>0. So the following computation applies to both x>1 and x<1. The expressions become quite long so we differentiate each term separately:

    ddx[xx21]=[x21+x2x21]=1x21[(x21)+x2]ddxlog|x+x21|=1x+x21[1+xx21]=1x+x21x+x21x21=1x21

    Putting things together then gives us

    ddx[12xx21+12log|x+x21|+C]=12x21[(x21)+x2+1]+0=x2x21

    This tells us that our answer for x>1 is also valid when x<1 and so

    x2x21dx=12xx21+12log|x+x21|+C

    when x<1 and when x>1.

In this example, we were lucky. The answer that we derived for x>1 happened to be also valid for x<1. This does not always happen with the x=asecu substitution. When it doesn't, we have to apply separate x>a and x<a analyses that are very similar to our x>1 analysis above. Of course that doubles the tedium. So in the CLP-2 problem book, we will not pose questions that require separate x>a and x<a computations.

The method, as we have demonstrated it above, works when our integrand contains the square root of very specific families of quadratic polynomials. In fact, the same method works for more general quadratic polynomials — all we need to do is complete the square 3.

Example 1.9.7 53x22x3x1dx

This time we have an integral with a square root in the integrand, but the argument of the square root, while a quadratic function of x, is not in one of the standard forms a2x2, a2+x2, x2a2. The reason that it is not in one of those forms is that the argument, x22x3, contains a term , namely 2x that is of degree one in x. So we try to manipulate it into one of the standard forms by completing the square.

Solution

  • We first rewrite the quadratic polynomial x22x3 in the form (xa)2+b for some constants a,b. The easiest way to do this is to expand both expressions and compare coefficients of x:

    x22x3=(xa)2+b=(x22ax+a2)+b

    So — if we choose 2a=2 (so the coefficients of x1 match) and a2+b=3 (so the coefficients of x0 match), then both expressions are equal. Hence we set a=1 and b=4. That is

    x22x3=(x1)24

    Many of you may have seen this method when learning to sketch parabolas.
  • Once this is done we can convert the square root of the integrand into a standard form by making the simple substitution y=x1. Here goes

    53x22x3x1dx=53(x1)24x1dx=42y24ydywith y=x1,dy=dx=π/304sec2u42secu 2secutanuduwith y=2secuand dy=2secutanudu

    Notice that we could also do this in fewer steps by setting (x1)=2secu,dx=2secutanudu.
  • To get the limits of integration we used that
    • the value of u that corresponds to y=2 obeys 2=y=2secu=2cosu or cosu=1, so that u=0 works and
    • the value of u that corresponds to y=4 obeys 4=y=2secu=2cosu or cosu=12, so that u=π3 works.
  • Now returning to the evaluation of the integral, we simplify and continue. 53x22x3x1dx=π/302sec2u1 tanudu=2π/30tan2udusince sec2u=1+tan2u

    In taking the square root of sec2u1=tan2u we used that tanu0 on the range 0uπ3.

    =2π/30[sec2u1]dusince sec2u=1+tan2u, again=2[tanuu]π/30=2[3π3]

Exercises

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

Stage 1
1 (✳)

For each of the following integrals, choose the substitution that is most beneficial for evaluating the integral.

  1. \displaystyle \int \frac{2x^2}{\sqrt{9x^2-16}} \, \, d{x}
  2. \displaystyle \int \frac{x^4-3}{\sqrt{1-4x^2}} \, \, d{x}
  3. \displaystyle \int {(25+x^2)}^{-5/2} \, \, d{x}
2

For each of the following integrals, choose a trigonometric substitution that will eliminate the roots.

  1. \displaystyle\int \dfrac{1}{\sqrt{x^2-4x+1}}\, d{x}
  2. \displaystyle\int \dfrac{(x-1)^6}{(-x^2+2x+4)^{3/2}}\, d{x}
  3. \displaystyle\int \dfrac{1}{\sqrt{4x^2+6x+10}}\, d{x}
  4. \displaystyle\int \sqrt{x^2-x}\, d{x}
3

In each part of this question, assume \theta is an angle in the interval \left[ 0,\pi/2\right]\text{.}

  1. If \sin\theta=\dfrac{1}{20}\text{,} what is \cos\theta ?
  2. If \tan\theta=7\text{,} what is \csc\theta ?
  3. If \sec\theta=\dfrac{\sqrt{x-1}}{2}\text{,} what is \tan\theta ?
4

Simplify the following expressions.

  1. \sin\left(\arccos \left(\frac{x}{2}\right)\right)
  2. \sin\left(\arctan \left(\frac{1}{\sqrt{3}}\right)\right)
  3. \sec\left(\arcsin \left(\sqrt{x}\right)\right)
Stage 2
5 (✳)

Evaluate \displaystyle\int \frac1{(x^2+4)^{3/2}} \,\, d{x}.

6 (✳)

Evaluate \displaystyle \int_{0}^{4}\frac{1}{(4+x^2)^{3/2}} \, d{x}\text{.} Your answer may not contain inverse trigonometric functions.

7 (✳)

Evaluate \displaystyle\int_0^{5/2} \frac{\, d{x}}{\sqrt{25-x^2}}\text{.}

8 (✳)

Evaluate \displaystyle\int \frac{\, d{x}}{\sqrt{x^2+25}}\text{.} You may use that {\displaystyle\int} \sec \, d{x} = \log\big|\sec x+\tan x\big|+C\text{.}

9

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

10 (✳)

Evaluate \displaystyle\int\frac{\, d{x}}{x^2\sqrt{x^2+16}}\text{.}

11 (✳)

Evaluate \displaystyle\int \frac{\, d{x}}{x^2\sqrt{x^2-9}} for x \ge 3\text{.} Do not include any inverse trigonometric functions in your answer.

12 (✳)

(a) Show that \displaystyle\int_0^{\pi/4}\cos^4\theta\, d{\theta}=(8+3\pi)/32\text{.}

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

13

Evaluate \displaystyle\int_{-\pi/12}^{\pi/12} \dfrac{15x^3}{(x^2+1)(9-x^2)^{5/2}}\, d{x}\text{.}

14 (✳)

Evaluate {\displaystyle\int} \sqrt{4-x^2}\,\, d{x}\text{.}

15 (✳)

Evaluate \displaystyle\int \frac{\sqrt{25x^2-4}}{x}\,\, d{x} for x\gt \frac{2}{5}\text{.}

16

Evaluate \displaystyle\int_{\sqrt{10}}^{\sqrt{17}} \frac{x^3}{\sqrt{x^2-1}}\, \, d{x}\text{.}

17 (✳)

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

18

Evaluate \displaystyle\int \dfrac{1}{(2x-3)^3\sqrt{4x^2-12x+8}}\, d{x} for x \gt 2\text{.}

19

Evaluate \displaystyle\int_0^1\dfrac{x^2}{(x^2+1)^{3/2}}\, d{x}\text{.}

You may use that \int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}

20

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

Stage 3
21

Evaluate \displaystyle\int \dfrac{x^2}{\sqrt{x^2-2x+2}}\, d{x}\text{.}

You may assume without proof that \displaystyle\int \sec^3 \theta\, d{\theta} = \frac{1}{2}\sec\theta\tan\theta + \frac{1}{2}\log|\sec\theta+\tan\theta|+C\text{.}

22

Evaluate \displaystyle\int \dfrac{1}{\sqrt{3x^2+5x}}\, d{x}\text{.}

You may use that \int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}

23

Evaluate \displaystyle\int\dfrac{(1+x^2)^{3/2}}{x}\, d{x}\text{.} You may use the fact that \displaystyle\int \csc \theta\, d{\theta}=\log|\cot \theta - \csc \theta|+C\text{.}

24

Below is the graph of the ellipse \left(\frac{x}{4}\right)^2+\left(\frac{y}{2}\right)^2=1\text{.} Find the area of the shaded region using the ideas from this section.

25

Let f(x) = \dfrac{|x|}{\sqrt[4]{1-x^2}}\text{,} and let R be the region between f(x) and the x-axis over the interval [-\frac{1}{2},\frac{1}{2}]\text{.}

  1. Find the area of R\text{.}
  2. Find the volume of the solid formed by rotating R about the x-axis.
26

Evaluate \displaystyle\int \sqrt{1+e^x}\, d{x}\text{.} You may use the antiderivative \displaystyle\int \csc \theta \, d{\theta} = \log|\cot \theta - \csc \theta|+C\text{.}

27

Consider the following work.

\begin{align*} \int \frac{1}{1-x^2}\, d{x}&=\int\dfrac{1}{1-\sin^2 \theta}\cos\theta\, d{\theta} \qquad \mbox{using } x=\sin\theta, \, d{x}=\cos\theta\, d{\theta}\\ &=\int \frac{\cos \theta}{\cos^2 \theta}\, d \theta\\ &=\int \sec \theta\, d{\theta}\\ &=\log|\sec \theta + \tan \theta| +C \qquad\qquad\qquad \mbox{Example }{\text{1.8.19}}\\ &=\log\left | \dfrac{1}{\sqrt{1-x^2}}+\dfrac{x}{\sqrt{1-x^2}} \right| +C \qquad \text{using the triangle below}\\ &=\log\left | \dfrac{1+x}{\sqrt{1-x^2}} \right| +C \end{align*}
 
 
  1. Differentiate \log\left| \dfrac{1+x}{\sqrt{1-x^2}}\right|\text{.}
  2. True or false: \displaystyle\int_{2}^{3} \frac{1}{1-x^2}\, d{x} = \left[\log\left| \dfrac{1+x}{\sqrt{1-x^2}}\right|\right]_{x=2}^{x=3}
  3. Was the work in the question correct? Explain.
28
  1. Suppose we are evaluating an integral that contains the term \sqrt{a^2-x^2}\text{,} where a is a positive constant, and we use the substitution x=a\sin u (with inverse u = \arcsin(x/a)), so that

    \sqrt{a^2-x^2} = \sqrt{a^2\cos^2u}= |a\cos u| \nonumber

    Under what circumstances is |a\cos u|\neq a\cos u\text{?}
  2. Suppose we are evaluating an integral that contains the term \sqrt{a^2+x^2}\text{,} where a is a positive constant, and we use the substitution x=a\tan u (with inverse u = \arctan(x/a)), so that

    \sqrt{a^2+x^2} = \sqrt{a^2\sec^2u}= |a\sec u| \nonumber

    Under what circumstances is |a\sec u|\neq a\sec u\text{?}
  3. Suppose we are evaluating an integral that contains the term \sqrt{x^2-a^2}\text{,} where a is a positive constant, and we use the substitution x=a\sec u (with inverse u = \textrm{arcsec}(x/a)=\arccos(a/x)), so that

    \sqrt{x^2-a^2} = \sqrt{a^2\tan^2u}= |a\tan u| \nonumber

    Under what circumstances is |a\tan u|\neq a\tan u\text{?}
  1. To paraphrase the Law of the Instrument, possibly Mark Twain and definitely some psychologists, when you have a shiny new hammer, everything looks like a nail.
  2. To be pedantic, we mean the range of the “standard” arctangent function or its “principle value”. One can define other arctangent functions with different ranges.
  3. If you have not heard of “completing the square” don't worry. It is not a difficult method and it will only take you a few moments to learn. It refers to rewriting a quadratic polynomial P(x) = ax^2 + bx + c as P(x)= a(x+d)^2 +e for new constants d,e\text{.}

 


This page titled 1.9: Trigonometric Substitution 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.

  • Was this article helpful?

Support Center

How can we help?