Question

$\int \frac{\cos 2x-1}{\cos 2x+1}dx=$
(a) tan x − x + C
(b) x + tan x + C
(c) x − tan x + C
(d) − x − cot x + C

Answer 1

(c) x − tan x + C

$$\begin{gathered} \int \left(\frac{\cos 2x-1}{\cos 2x+1}\right)dx \\ =\int \left(\frac{1-2{\sin }^{2}x-1}{2{\cos }^{2}x-1+1}\right)dx \\ =-\int {\tan }^{2}xdx \\ =-\int \left({\sec }^{2}x-1\right)dx \\ =\int \left(1-{\sec }^{2}x\right)dx \\ =x-\tan x+C \end{gathered}$$