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.
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
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.
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.
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
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
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