009B Sample Midterm 2, Problem 5

Evaluate the integral:

$\int \tan ^{4}x~dx$

Foundations:
Trig identity
U substitution

Solution:

Step 1:
First, we write $\int \tan ^{4}(x)dx=\int \tan ^{2}(x)\tan ^{2}(x)dx$.
Using the trig identity $\sec ^{2}(x)=\tan ^{2}(x)+1$, we have $\tan ^{2}(x)=\sec ^{2}(x)-1$.
Plugging in the last identity into one of the $\tan ^{2}(x)$, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \tan ^{4}(x)dx=\int \tan ^{2}(x)(\sec ^{2}(x)-1)dx=\int \tan ^{2}(x)\sec ^{2}(x)dx-\int \tan ^{2}(x)dx=\int \tan ^{2}(x)\sec ^{2}(x)dx-\int (\sec ^{2}x-1)dx}
using the identity again on the last equality

Step 2:
So, we have $\int \tan ^{4}(x)dx=\int \tan ^{2}(x)\sec ^{2}(x)dx-\int (\sec ^{2}x-1)dx$.
For the first integral, we need to use substitution. Let $u=\tan(x)$. Then, $du=\sec ^{2}(x)dx$.
So, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \tan ^{4}(x)dx=\int u^{2}du-\int (\sec ^{2}(x)-1)dx} .

Step 3:
We integrate to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \tan ^{4}(x)dx={\frac {u^{3}}{3}}-(\tan(x)-x)+C={\frac {\tan ^{3}(x)}{3}}-\tan(x)+x+C}

Final Answer:
${\frac {\tan ^{3}(x)}{3}}-\tan(x)+x+C$

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