009B Sample Midterm 1, Problem 5 Detailed Solution

Evaluate the integral:

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

Foundations:
1. Recall the trig identity
$\sin ^{2}x+\cos ^{2}x=1$
2. How would you integrate  $\int \sin ^{2}x\cos x~dx?$
You can use  $u$-substitution.
Let  $u=\sin x.$
Then,  $du=\cos x~dx.$
Thus,
${\begin{array}{rcl}\displaystyle {\int \sin ^{2}x\cos x~dx}&=&\displaystyle {\int u^{2}~du}\\&&\\&=&\displaystyle {{\frac {u^{3}}{3}}+C}\\&&\\&=&\displaystyle {{\frac {\sin ^{3}x}{3}}+C.}\end{array}}$

Solution:

Step 1:
First, we write
$\int \sin ^{3}x\cos ^{2}x~dx=\int (\sin x)\sin ^{2}x\cos ^{2}x~dx.$
Using the identity  $\sin ^{2}x+\cos ^{2}x=1,$
we get
$\sin ^{2}x=1-\cos ^{2}x.$
If we use this identity, we have
${\begin{array}{rcl}\displaystyle {\int \sin ^{3}x\cos ^{2}x~dx}&=&\displaystyle {\int (\sin x)(1-\cos ^{2}x)\cos ^{2}x~dx}\\&&\\&=&\displaystyle {\int (\cos ^{2}x-\cos ^{4}x)\sin(x)~dx.}\end{array}}$

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

Final Answer:
${\frac {\cos ^{5}x}{5}}-{\frac {\cos ^{3}x}{3}}+C$

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