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Mathematics LibreTexts

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=uvvdu

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=uvvdu

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.

Example 1

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=uvvduxsinxdx=x(cosx)(sinx+C1)xsinxdx=xcosx+sinx+C

Example 2

Evaluate arcsinxdx. A choice of u and dv which works is

Solution

u=arcsinx,dv=dx

We may take v=x. Then

du=dx1x2vdu=xdx1x2=1x2+C1

Finally, arcsinxdx=xarcsinx(1x2+C1),

arcsinxdx=xarcsinx+1x2+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.

Example 3

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=uvvdu=x2cosx+2xcosxdx

Step 2 Evaluate 2xcosxdx.

u1=2x,dv1=cosxdxdu1=2dx,dv1=cosxdx=sinx+C

We take v1=sinx.

2xcosxdx=u1v1v1du1=2xsinx2sinxdx=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.

Example 4

Evaluate sin2θdθ. Let

Solution

Then

u=sinθ,dv=sinθdθdu=cosθdθ,v=cosθ

sin2θdθ=sinθcosθcos2θdθ=sinθcosθ+cos2θdθ=sinθcosθ+(1sin2θ)dθ=sinθcosθ+θsin2θdθ

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=1cos(20)2

This is derived from the addition formula,

cos(θ+ϕ)=cosθcosϕsinθsinϕcos(2θ)=cos2θsin2θ=12sin2θsin2θ=1cos(2θ)2

Then

sin2θdθ=1cos2θ2dθ=12dθ12cos2θdθ=12dθ14cos2θd(2θ)=12θ14sin2θ+C

This answer agrees with Example 4 because

so

sin2θ=sin(θ+θ)=2sinθcosθ12θ14sin2θ=12θ12sinθcosθ

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)]babag(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)

image
Figure 7.4.1 Definite Integration by Parts
Example 5

Evaluate π0xsinxdx (Figure 7.4.2). Take u=x,dv=sinxdx as in Example 1. Then v=cosx and

π0xsinxdx=xcosx]π0π0cosxdx=xcosx]π0+sinx]π0=(π(1)+01)+(00)=π

Solution

 

 

image
Figure 7.4.2

 

PROBLEMS FOR SECTION 7.4

Evaluate the integrals in Problems 1-35.
1xcosxdx
2arccosxdx
3t2costdt
4xarctanxdx
5tsin(2t1)dt
6arcsin(3t)dt
7x2sin(4x)dx
8xarcsecxdx
9x3arcsecxdx
10x3sinxdx
11sinxdx
12sinθtan2θdθ
13arctanxdx
14xtanxsec2xdx
15
x3x21dx
16cos2θdθ
17xsinxcosxdx
18tsin2tdt
19
sinθsin(2θ)dθ
20 cosxcos(3x)dx
21sinxcos(5x)dx
22cosxcot4xdx
23t3sin(t2)dt
24x3cos(2x21)dx
25
1x3sin(1x)dx
26
sinθcosθcos(sinθ)dθ
27t3t2+4dt
281x31x1dx
29
π20θcosθdθ
301.20arcsinxdx
31π0sin2θdθ
3210arcsinxdx
33
10xarccotxdx
34x0xarccotxdx

36 Find the volume of the solid of revolution generated by rotating the region under the curve y=sinx,0xπ, 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=u2v2udu

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


This page titled 7.4: Integration by Parts is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by H. Jerome Keisler.

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