7.4: Integration by Parts
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One reason it is harder to integrate than differentiate is that for derivatives there is both a Sum Rule and a Product Rule,
d(u+v)=du+dv,d(uv)=udv+vdu
while for integrals there is only a Sum Rule,
∫du+dv=∫du+∫dv
The Sum Rule for integrals is obtained in a simple way by reversing the sum rule for derivatives.
There is a way to turn the Product Rule for derivatives into a rule for integrals. It no longer looks like a product rule, and is called integration by parts. Integration by parts is a basic method which is needed for many integrals involving trigonometric functions (and later exponential functions).
Integration by Parts
Suppose, for x in an open interval I, that u and v depend on x and that du and dv exist. Then
∫udv=uv−∫vdu
PROOF We use the Product Rule
udv+vdu=d(uv),udv=d(uv)−vdu
Integrating both sides with x as the independent variable,
∫udv=∫(d(uv)−vdu)=∫d(uv)−∫vdu=uv−∫vdu
No constant of integration is needed because there are indefinite integrals on both sides of the equation.
Integration by parts is useful whenever ∫vdu is easier to evaluate than a given integral ∫udv.
Evaluate ∫xsinxdx. Our plan is to break xsinxdx into a product of the form udv, evaluate the integrals ∫dv and ∫vdu, and then use integration by parts to get ∫udc. There are several choices we might make for u and di, and not all of them lead to a solution of the problem. Some guesswork is required.
Solution
First try:u=sinx,dv=xdx.∫dv=∫xdx=12x2+C. Take v=12x2. Next we find du and try to evaluate ∫vdu.
du=cosxdx,∫vdu=∫12x2cosxdx
This integral looks harder than the one we started with, so we shall start over with another choice of u and dv.
Second try: u=x,du=sinxdx.
∫dv=∫sinxdx=−cosx+C
We take v=−cosx. This time we find du and easily evaluate ∫vdu.
du=dx,∫udu=∫−cosxdx=−sinx+C1
Finally we use the rule
∫udv=uv−∫vdu∫xsinxdx=x(−cosx)−(−sinx+C1)∫xsinxdx=−xcosx+sinx+C
Evaluate ∫arcsinxdx. A choice of u and dv which works is
Solution
u=arcsinx,dv=dx
We may take v=x. Then
du=dx√1−x2∫vdu=∫−xdx√1−x2=−√1−x2+C1
Finally, ∫arcsinxdx=xarcsinx−(−√1−x2+C1),
∫arcsinxdx=xarcsinx+√1−x2+C
This integral and the similar formula for ∫arccosxdx are included in our table at the end of the book. We shall see how to integrate the other inverse trigonometric functions in the next chapter.
Evaluate ∫x2sinxdx. This requires two integrations by parts.
Solution
Step 1
u=x2,dv=sinxdxdu=2xdx,∫dv=∫sinxdx=−cosx+C
We take v=−cosx.
∫x2sinxdx=uv−∫vdu=−x2cosx+∫2xcosxdx
Step 2 Evaluate ∫2xcosxdx.
u1=2x,dv1=cosxdxdu1=2dx,∫dv1=∫cosxdx=sinx+C
We take v1=sinx.
∫2xcosxdx=u1v1−∫v1du1=2xsinx−∫2sinxdx=2xsinx+2cosx+C.
Combining the two steps,
∫x2sinxdx=−x2cosx+2xsinx+2cosx+C
Sometimes integration by parts will yield an equation in which the given integral occurs on both sides. One can often solve for the answer.
Evaluate ∫sin2θdθ. Let
Solution
Then
u=sinθ,dv=sinθdθdu=cosθdθ,v=−cosθ
∫sin2θdθ=−sinθcosθ−∫−cos2θdθ=−sinθcosθ+∫cos2θdθ=−sinθcosθ+∫(1−sin2θ)dθ=−sinθcosθ+θ−∫sin2θdθ
We solve this equation for ∫sin2θdθ,
∫sin2θdθ=−12sinθcosθ+12θ+C
Here is another way to evaluate ∫sin2θdθ. Instead of using integration by parts, we can use the half-angle formula
sin2θ=1−cos(20)2
This is derived from the addition formula,
cos(θ+ϕ)=cosθcosϕ−sinθsinϕcos(2θ)=cos2θ−sin2θ=1−2sin2θsin2θ=1−cos(2θ)2
Then
∫sin2θdθ=∫1−cos2θ2dθ=12∫dθ−12∫cos2θdθ=12∫dθ−14∫cos2θd(2θ)=12θ−14sin2θ+C
This answer agrees with Example 4 because
so
sin2θ=sin(θ+θ)=2sinθcosθ12θ−14sin2θ=12θ−12sinθcosθ
Integration by parts requires a great deal of guesswork. Given a problem ∫h(x)dx we try to find a way to split h(x)dx into a product f(x)g′(x)dx where we can evaluate both of the integrals ∫g′(x)dx and ∫g(x)f′(x)dx.
Definite integrals take the following form when integration by parts is applied.
DEFINITE INTEGRATION BY PARTS
If u=f(x) and x=g(x) have continuous derivatives on an open interval I, then for a,b in I,
∫baf(x)g′(x)dx=f(x)g(x)]ba−∫bag(x)f′(x)dx
PROOF The Product Rule gives
f(x)g′(x)dx+g(x)f′(x)dx=d(f(x)g(x))
Then by the Fundamental Theorem of Calculus,
∫ba(f(x)g′(x)+g(x)f′(x))dx=f(x)g(x)]ba
and the desired result follows by the Sum Rule.
If we plot u=f(x) on one axis and v=g(x) on the other, we get a picture of definite integration by parts (Figure 7.4.1). The picture is easier to interpret if we change variables in the definite integrals and write the formula for integration by parts in the form
∫g(b)g(a)udv+∫f(b)f(a)vdu=f(b)g(b)−f(a)g(a)
Evaluate ∫π0xsinxdx (Figure 7.4.2). Take u=x,dv=sinxdx as in Example 1. Then v=−cosx and
∫π0xsinxdx=−xcosx]π0−∫π0−cosxdx=−xcosx]π0+sinx]π0=(−π(−1)+0⋅1)+(0−0)=π
Solution
PROBLEMS FOR SECTION 7.4
Evaluate the integrals in Problems 1-35.
1∫xcosxdx
2∫arccosxdx
3∫t2costdt
4∫xarctanxdx
5∫tsin(2t−1)dt
6∫arcsin(3t)dt
7∫x2sin(4x)dx
8∫xarcsecxdx
9∫x3arcsecxdx
10∫x3sinxdx
11∫sin√xdx
12∫sinθtan2θdθ
13∫arctanxdx
14∫xtanxsec2xdx
15
∫x3√x2−1dx
16∫cos2θdθ
17∫xsinxcosxdx
18∫tsin2tdt
19
∫sinθsin(2θ)dθ
20 ∫cosxcos(3x)dx
21∫sinxcos(5x)dx
22∫cosxcot4xdx
23∫t3sin(t2)dt
24∫x3cos(2x2−1)dx
25
∫1x3sin(1x)dx
26
∫sinθcosθcos(sinθ)dθ
27∫t3√t2+4dt
28∫1x3√1x−1dx
29
∫π⋅20θcosθdθ
30∫1.20arcsinxdx
31∫π0sin2θdθ
32∫10arcsinxdx
33
∫10xarccotxdx
34∫x0xarccotxdx
36 Find the volume of the solid of revolution generated by rotating the region under the curve y=sinx,0≤x≤π, about (a) the x-axis, (b) the y-axis.
37 Prove that if f is a differentiable function of x, then
∫f(x)dx=xf(x)−∫xf′(x)dx
38 If u and v are differentiable functions of x, show that
∫u2du=u2v−2∫udu
39 Show that if f′ and g are differentiable for all x, then
∫g(x)g′(x)f′′(g(x))dx=f′(g(x))g(x)−f(g(x))+C