Question

Evaluate: $\int \frac{x}{1+\cos 2x}\ dx$

Answer 1

Solution:

As we know that

$1+ cos2x= 2 cos^{2}x$

So substitute this value in the equation

$\int \frac{x}{1+ cos2x} dx= \int \frac{x}{2 cos^{2}x} dx\\\\=\frac{1}{2}\int x sec^{2}x dx$

Now this form can be integrate by parts,assuming trigonometric function as first function

$\int x sec^{2}x dx=x\int sec^{2}x dx-\int( \frac{dx}{dx}\int sec^{2}x dx)dx \\\\= x\: tan x-\int tan\: x dx\\\\=x\: tan\:x +log cos\: x+C$

so,finally put values in the equation

$\int \frac{x}{1+ cos\:2x} dx=\frac{1}{2}[x\: tan\:x +log\: cos\: x]+C$