009B Sample Midterm 3, Problem 5

Evaluate the indefinite and definite integrals.

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 \int \tan^3x ~dx}

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 \int_0^\pi \sin^2x~dx}

Foundations:
Review u substitution
Trig identities

Solution:

(a)

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

Step 2:

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