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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \tan ^{3}xdx=\int \tan ^{2}x\tan xdx} .
Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \tan ^{2}x=\sec ^{2}x-1} , we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u=\tan(x)} . Then, $du=\sec ^{2}xdx$. So, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \tan ^{3}xdx={\frac {\tan ^{2}x}{2}}-\int {\frac {\sin x}{\cos x}}dx} .
Now, we let $u=\cos x$. Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle du=-\sin xdx} . So, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:
One of the double angle formulas is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos(2x)=1-2\sin ^{2}(x)} . Solving for $\sin ^{2}(x)$, we get $\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$.
Plugging this identity into our integral, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{0}^{\pi }\sin ^{2}x~dx=\int _{0}^{\pi }{\frac {1-\cos(2x)}{2}}dx=\int _{0}^{\pi }{\frac {1}{2}}dx-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}dx} .

Step 2:
If we integrate the first integral, we get
$\int _{0}^{\pi }\sin ^{2}x~dx=\left.{\frac {x}{2}}\right|_{0}^{\pi }-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}dx={\frac {\pi }{2}}-\int _{0}^{\pi }{\frac {\cos(2x)}{2}}dx$.

Step 3:
For the remaining integral, we need to use u substitution. Let $u=2x$. Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle du=2dx} . Also, since this is a definite integral
and we are using u substiution, we need to change the bounds of integration. We have $u_{1}=2(0)=0$ 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 u_2=2(\pi)=2\pi} .
So, the integral becomes
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_0^\pi \sin^2x~dx=\frac{\pi}{2}-\int_0^{2\pi} \frac{\cos(u)}{4}du=\frac{\pi}{2}-\left.\frac{\sin(u)}{4}\right|_0^{2\pi}=\frac{\pi}{2}-\bigg(\frac{\sin(2\pi)}{4}-\frac{\sin(0)}{4}\bigg)=\frac{\pi}{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 \frac{\tan^2x}{2}+\ln |\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{\pi}{2}} Return to Sample Exam