009B Sample Final 3, Problem 5

Find the following integrals.

(a)  $\int x\cos(x)~dx$

(b)  $\int \sin ^{3}(x)\cos ^{2}(x)~dx$

Foundations:
1. Integration by parts tells us that
$\int u~dv=uv-\int v~du.$
2. Since  $\sin ^{2}x+\cos ^{2}x=1,$  we have
$\sin ^{2}x=1-\cos ^{2}x.$

Solution:

(a)

Step 1:
To calculate this integral, we use integration by parts.
Let  $u=x$  and  $dv=\cos xdx.$
Then,  $du=dx$  and  $v=\sin x.$
Therefore, we have
$\int x\cos(x)~dx=x\sin x-\int \sin x~dx.$

Step 2:
Then, we integrate to get
$\int x\cos(x)~dx=x\sin x+\cos x+C.$

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)    $x\sin x+\cos x+C$
(b)
(c)

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