Evaluate `int tan^(-1)(secx+tanx)dx,-pi//2ltxlt pi//2`

Explanation

Solution:

`\int \tan ^{-1}(\sec x+\tan x) d x=\int \tan ^{-1}(\frac{1}{\cos x}+\frac{\sin x}{\cos x}) d x`

=`\int \tan ^{-1}(\frac{1+\sin x}{\cos x}) d x=\int \tan ^{-1}\{\frac{1-\cos (\frac{\pi}{2}+x)}{\sin (\frac{\pi}{2}+x)}\} d x`

=`\int \tan ^{-1}\{\frac{2 \sin ^{2}(\frac{\pi}{4}+\frac{x}{2})}{2 \sin (\frac{\pi}{4}+\frac{x}{2}) \cos (\frac{\pi}{4}+\frac{x}{2})}\} d x`

=`\int \tan ^{-1}\{\tan (\frac{\pi}{4}+\frac{x}{2})\} d x=\int(\frac{\pi}{4}+\frac{x}{2}) d x`

=`\frac{\pi}{4} \cdot \int d x+\frac{1}{2} \int x d x=\frac{\pi x}{4}+\frac{1}{4} x^{2}+C`