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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)=d u+d v, \quad d(u v)=u d v+v d u \nonumber

while for integrals there is only a Sum Rule,

\int d u+d v=\int d u+\int d v \nonumber

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 d v exist. Then

\int u d v=u v-\int v d u \nonumber

PROOF We use the Product Rule

u d v+v d u=d(u v), \quad u d v=d(u v)-v d u \nonumber

Integrating both sides with x as the independent variable,

\int u d v=\int(d(u v)-v d u)=\int d(u v)-\int v d u=u v-\int v d u \nonumber

No constant of integration is needed because there are indefinite integrals on both sides of the equation.

Integration by parts is useful whenever \int v d u is easier to evaluate than a given integral \int u d v.

Example 1

Evaluate \int x \sin x d x. Our plan is to break x \sin x d x into a product of the form u d v, evaluate the integrals \int d v and \int v d u, and then use integration by parts to get \int u d c. There are several choices we might make for u and d i, and not all of them lead to a solution of the problem. Some guesswork is required.

Solution

First \operatorname{tr} y: u=\sin x, d v=x d x . \int d v=\int x d x=\frac{1}{2} x^{2}+C. Take v=\frac{1}{2} x^{2}. Next we find d u and try to evaluate \int v d u.

d u=\cos x d x, \quad \int v d u=\int \frac{1}{2} x^{2} \cos x d x \nonumber

This integral looks harder than the one we started with, so we shall start over with another choice of u and d v.

Second try: u=x, d u=\sin x d x.

\int d v=\int \sin x d x=-\cos x+C \nonumber

We take v=-\cos x. This time we find d u and easily evaluate \int v d u.

d u=d x, \quad \int u d u=\int-\cos x d x=-\sin x+C_{1} \nonumber

Finally we use the rule

\begin{aligned} \int u d v & =u v-\int v d u \\ \int x \sin x d x & =x(-\cos x)-\left(-\sin x+C_{1}\right) \\ \int x \sin x d x & =-x \cos x+\sin x+C \end{aligned} \nonumber

Example 2

Evaluate \int \arcsin x d x. A choice of u and d v which works is

Solution

u=\arcsin x, \quad d v=d x \nonumber

We may take v=x. Then

\begin{aligned} d u & =\frac{d x}{\sqrt{1-x^{2}}} \\ \int v d u & =\int-\frac{x d x}{\sqrt{1-x^{2}}}=-\sqrt{1-x^{2}}+C_{1} \end{aligned} \nonumber

Finally, \quad \int \arcsin x d x=x \arcsin x-\left(-\sqrt{1-x^{2}}+C_{1}\right),

\int \arcsin x d x=x \arcsin x+\sqrt{1-x^{2}}+C \nonumber

This integral and the similar formula for \int \arccos x d x 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 \int x^{2} \sin x d x. This requires two integrations by parts.

Solution

Step 1

\begin{aligned} u & =x^{2}, & d v & =\sin x d x \\ d u & =2 x d x, & \int d v & =\int \sin x d x=-\cos x+C \end{aligned} \nonumber

We take v=-\cos x.

\int x^{2} \sin x d x=u v-\int v d u=-x^{2} \cos x+\int 2 x \cos x d x \nonumber

Step 2 Evaluate \int 2 x \cos x d x.

\begin{array}{rlrl} u_{1} & =2 x, & d v_{1} & =\cos x d x \\ d u_{1} & =2 d x, \quad \int d v_{1} & =\int \cos x d x=\sin x+C \end{array} \nonumber

We take v_{1}=\sin x.

\begin{aligned} \int 2 x \cos x d x & =u_{1} v_{1}-\int v_{1} d u_{1} \\ & =2 x \sin x-\int 2 \sin x d x \\ & =2 x \sin x+2 \cos x+C . \end{aligned} \nonumber

Combining the two steps,

\int x^{2} \sin x d x=-x^{2} \cos x+2 x \sin x+2 \cos x+C \nonumber

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 \int \sin ^{2} \theta d \theta. Let

Solution

Then

\begin{aligned} u & =\sin \theta, & d v & =\sin \theta d \theta \\ d u & =\cos \theta d \theta, & v & =-\cos \theta \end{aligned} \nonumber

\begin{aligned} \int \sin ^{2} \theta d \theta & =-\sin \theta \cos \theta-\int-\cos ^{2} \theta d \theta \\ & =-\sin \theta \cos \theta+\int \cos ^{2} \theta d \theta \\ & =-\sin \theta \cos \theta+\int\left(1-\sin ^{2} \theta\right) d \theta \\ & =-\sin \theta \cos \theta+\theta-\int \sin ^{2} \theta d \theta \end{aligned} \nonumber

We solve this equation for \int \sin ^{2} \theta d \theta,

\int \sin ^{2} \theta d \theta=-\frac{1}{2} \sin \theta \cos \theta+\frac{1}{2} \theta+C \nonumber

Here is another way to evaluate \int \sin ^{2} \theta d \theta. Instead of using integration by parts, we can use the half-angle formula

\sin ^{2} \theta=\frac{1-\cos (20)}{2} \nonumber

This is derived from the addition formula,

\begin{aligned} \cos (\theta+\phi) & =\cos \theta \cos \phi-\sin \theta \sin \phi \\ \cos (2 \theta) & =\cos ^{2} \theta-\sin ^{2} \theta=1-2 \sin ^{2} \theta \\ \sin ^{2} \theta & =\frac{1-\cos (2 \theta)}{2} \end{aligned} \nonumber

Then

\begin{aligned} \int \sin ^{2} \theta d \theta & =\int \frac{1-\cos 2 \theta}{2} d \theta=\frac{1}{2} \int d \theta-\frac{1}{2} \int \cos 2 \theta d \theta \\ & =\frac{1}{2} \int d \theta-\frac{1}{4} \int \cos 2 \theta d(2 \theta)=\frac{1}{2} \theta-\frac{1}{4} \sin 2 \theta+C \end{aligned} \nonumber

This answer agrees with Example 4 because

so

\begin{aligned} & \sin 2 \theta=\sin (\theta+\theta)=2 \sin \theta \cos \theta \\ & \frac{1}{2} \theta-\frac{1}{4} \sin 2 \theta=\frac{1}{2} \theta-\frac{1}{2} \sin \theta \cos \theta \end{aligned} \nonumber

Integration by parts requires a great deal of guesswork. Given a problem \int h(x) d x we try to find a way to split h(x) d x into a product f(x) g^{\prime}(x) d x where we can evaluate both of the integrals \int g^{\prime}(x) d x and \int g(x) f^{\prime}(x) d x.

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,

\left.\int_{a}^{b} f(x) g^{\prime}(x) d x=f(x) g(x)\right]_{a}^{b}-\int_{a}^{b} g(x) f^{\prime}(x) d x \nonumber

PROOF The Product Rule gives

f(x) g^{\prime}(x) d x+g(x) f^{\prime}(x) d x=d(f(x) g(x)) \nonumber

Then by the Fundamental Theorem of Calculus,

\left.\int_{a}^{b}\left(f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right) d x=f(x) g(x)\right]_{a}^{b} \nonumber

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

\int_{g(a)}^{g(b)} u d v+\int_{f(a)}^{f(b)} v d u=f(b) g(b)-f(a) g(a) \nonumber

image
Figure 7.4.1 Definite Integration by Parts
Example 5

Evaluate \int_{0}^{\pi} x \sin x d x (Figure 7.4.2). Take u=x, d v=\sin x d x as in Example 1. Then v=-\cos x and

\begin{aligned} \int_{0}^{\pi} x \sin x d x & =-x \cos x]_{0}^{\pi}-\int_{0}^{\pi}-\cos x d x \\ & \left.=-x \cos x]_{0}^{\pi}+\sin x\right]_{0}^{\pi} \\ & =(-\pi(-1)+0 \cdot 1)+(0-0)=\pi \end{aligned} \nonumber

Solution

 

 

image
Figure 7.4.2

 

PROBLEMS FOR SECTION 7.4

Evaluate the integrals in Problems 1-35.
1 \int x \cos x d x
2 \int \arccos x d x
3 \int t^{2} \cos t d t
4 \int x \arctan x d x
5 \int t \sin (2 t-1) d t
6 \int \arcsin (3 t) d t
7 \quad \int x^{2} \sin (4 x) d x
8 \quad \int x \operatorname{arcsec} x d x
9 \int x^{3} \operatorname{arcsec} x d x
10 \int x^{3} \sin x d x
11 \int \sin \sqrt{x} d x
12 \int \sin \theta \tan ^{2} \theta d \theta
13 \int \arctan x d x
14 \int x \tan x \sec ^{2} x d x
15
\int \frac{x^{3}}{\sqrt{x^{2}-1}} d x
16 \int \cos ^{2} \theta d \theta
17 \int x \sin x \cos x d x
18 \int t \sin ^{2} t d t
19
\int \sin \theta \sin (2 \theta) d \theta
20 \int \cos x \cos (3 x) d x
21 \int \sin x \cos (5 x) d x
22 \int \cos x \cot ^{4} x d x
23 \int t^{3} \sin \left(t^{2}\right) d t
24 \int x^{3} \cos \left(2 x^{2}-1\right) d x
25
\int \frac{1}{x^{3}} \sin \left(\frac{1}{x}\right) d x
26
\int \sin \theta \cos \theta \cos (\sin \theta) d \theta
27 \int t^{3} \sqrt{t^{2}+4} d t
28 \int \frac{1}{x^{3}} \sqrt{\frac{1}{x}-1} d x
29
\int_{0}^{\pi \cdot 2} \theta \cos \theta d \theta
30 \quad \int_{0}^{1.2} \arcsin x d x
31 \int_{0}^{\pi} \sin ^{2} \theta d \theta
32 \int_{0}^{1} \arcsin x d x
33
\int_{0}^{1} x \operatorname{arccot} x d x
34 \int_{0}^{x} x \operatorname{arccot} x d x

36 Find the volume of the solid of revolution generated by rotating the region under the curve y=\sin x, 0 \leq x \leq \pi, about (a) the x-axis, (b) the y-axis.

37 Prove that if f is a differentiable function of x, then

\int f(x) d x=x f(x)-\int x f^{\prime}(x) d x \nonumber

38 If u and v are differentiable functions of x, show that

\int u^{2} d u=u^{2} v-2 \int u d u \nonumber

39 Show that if f^{\prime} and g are differentiable for all x, then

\int g(x) g^{\prime}(x) f^{\prime \prime}(g(x)) d x=f^{\prime}(g(x)) g(x)-f(g(x))+C \nonumber


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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