009B Sample Midterm 3, Problem 5

Evaluate the indefinite and definite integrals.

a) $\int \tan ^{3}x~dx$

b) $\int _{0}^{\pi }\sin ^{2}x~dx$

Foundations:
Review u substitution
Trig identities

Solution:

(a)

Step 1:
We start by writing $\int \tan ^{3}xdx=\int \tan ^{2}x\tan xdx$.
Since $\tan ^{2}x=\sec ^{2}x-1$, we have $\int \tan ^{3}xdx=\int (\sec ^{2}x-1)\tan xdx=\int \sec ^{2}\tan xdx-\int \tan xdx$.

Step 2:
Now, we need to use u substitution for the first integral. Let $u=\tan(x)$. Then, $du=\sec ^{2}xdx$. So, we have
$\int \tan ^{3}xdx=\int udu-\int \tan xdx={\frac {u^{2}}{2}}-\int \tan xdx={\frac {\tan ^{2}x}{2}}-\int \tan xdx$.

Step 3:
For the remaining integral, we need to use u substitution. First, we write $\int \tan ^{3}xdx={\frac {\tan ^{2}x}{2}}-\int {\frac {\sin x}{\cos x}}dx$.
Now, we let $u=\cos x$. Then, $du=-\sin xdx$. So, we get
$\int \tan ^{3}xdx={\frac {\tan ^{2}x}{2}}+\int {\frac {1}{u}}dx={\frac {\tan ^{2}x}{2}}+\ln |u|+C={\frac {\tan ^{2}x}{2}}+\ln |\cos x|+C$.

(b)

Step 1:

Step 2:

Final Answer:
(a) ${\frac {\tan ^{2}x}{2}}+\ln |\cos x|+C$
(b) Return to Sample Exam