Question

Evaluate ${\int }_0^{\pi }\frac{x}{1+\sin x}dx$.

• A. $x\tan x-ln|\cos x|-x\sec x-ln|\sec x-\tan x|+C$
• B. $x\tan x+ln|\cos x|-x\sec x-ln|\sec x-\tan x|+C$
• C. $\tan x+ln|\cos x|-x\sec x-ln|\sec x-\tan x|+C$
• D. None of these

Answer 1

The correct option is B $x\tan x+ln|\cos x|-x\sec x-ln|\sec x-\tan x|+C$

$\int \frac{x(1-\sin x)}{1-{\sin }^{2}x}dx$

$\int \frac{x(1-\sin x)}{{\cos }^{2}x}dx$

$\int (x{\sec }^{2}x-x\tan x\sec x)dx$

Applying byparts,we get

$x\tan x+ln|\cos x|-x\sec x-ln|\sec x-\tan x|+C$