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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=\sin x.}
Therefore, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x\cos(x)~dx=x\sin x -\int \sin x~dx.}

Step 2:
Then, we integrate to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x\cos(x)~dx=x\sin x +\cos x+C.}

(b)

Step 1:

Step 2:

Final Answer:
(a)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\sin x +\cos x+C}
(b)

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