Evaluate the following integrals:


Evaluate the following integrals:

$\int x \tan ^{2} x d x$


Let $I=\int x \tan ^{2} x d x$

$=\int x\left(\sec ^{2} x-1\right) d x$

$=\int x \sec ^{2} x d x-\int x d x$

Using integration by parts,

$=x \int \sec ^{2} x d x-\int \frac{d}{d x} x \int \sec ^{2} x d x-\frac{x^{2}}{2}$

We know that, $\int \sec ^{2} x d x=\tan x$

$=x \tan x-\int \tan x d x-\frac{x^{2}}{2}$

$=x \tan x-\log |\sec x|-\frac{x^{2}}{2}+c$

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