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:
First, we use the identity  $\sin ^{2}x=1-\cos ^{2}x$  to get
${\begin{array}{rcl}\displaystyle {\int \sin ^{3}(x)\cos ^{2}(x)~dx}&=&\displaystyle {\int \sin ^{2}x(\cos ^{2}x)\sin x~dx}\\&&\\&=&\displaystyle {\int (1-\cos ^{2}x)(\cos ^{2}x)\sin x~dx.}\end{array}}$

Step 2:
Now, we use  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 u} -substitution.
Let  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 u=\cos(x).}   Then,  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 du=-\sin(x)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 -du=\sin(x)dx.}
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 \begin{array}{rcl} \displaystyle{\int \sin^3(x)\cos^2(x)~dx} & = & \displaystyle{\int (-1)(1-u^2)u^2~du}\\ &&\\ & = & \displaystyle{\int u^4-u^2~du}\\ &&\\ & = & \displaystyle{\frac{u^5}{5}-\frac{u^3}{3}+C}\\ &&\\ & = & \displaystyle{\frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C.} \end{array}}

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)    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 \frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C}

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