Question

Integrate : $\int \frac{1+\tan x}{x+\log \sec x}dx$

Answer 1

Given : $\int \frac{1+\tan x}{x+\log \sec x}dx$

Let $I=\int \frac{1+\tan x}{x+\log \sec x}dx$

Let $t=x+\log \sec x$

$\Rightarrow dt=(1+\frac{1}{\sec x}\sec x\tan x)dx$

$\Rightarrow dt=(1+\tan x)dx$

$\int \frac{(1+\tan x)dx}{x+\log \sec x}$$=\int \frac{dt}{t}$

$I=\log \left|t\right|+c$

where $(t=x+\log \sec x)$

$I=\log |x+\log \sec x|+c$