Question

$\int\frac{2x+\sin2x}{1+\cos2x}\, dx$ is equal to

• A. $x +\log |\tan x| + C$
• B. $x \log |\tan x| + C$
• C. $x \tan x + C$
• D. $\log |\cos x| + C$
• E. $\log |\sin x| + C$

Answer 1

Answer: (C)

Let $ I=\int \frac{2 x}{1+\cos 2 x} $
$=\int \frac{2 x}{1+\cos 2 x}+\frac{\sin 2 x}{1+\cos 2 x} d x $
$=\int \frac{2 x}{2 \cos ^{2} x}+\int \tan x d x $
$=\int x \sec ^{2} x+\int \tan x d x $
$=x \tan x-\int \tan x d x+\int \tan x d x $
$=x \tan x+C $