Question

Evaluate $\int \frac{(1-\cos x)dx}{\cos x(1+\cos x)}$

• A. $\log (\sec x+\tan x)-2\tan \frac{x}{2}+c$
• B. $\log (\sec x+\tan x)+2\tan \frac{x}{2}+c$
• C. $\log (\sec x-\tan x)-2\tan \frac{x}{2}+c$
• D. $\log (\sec x-\tan x)+2\tan \frac{x}{2}+c$

Answer 1

The correct option is A $\log (\sec x+\tan x)-2\tan \frac{x}{2}+c$

Given, $\int \frac{(1-\cos x)dx}{\cos x(1+\cos x)}$

$\Rightarrow$ $\int \frac{(1+\cos x)}{\cos x(1+\cos x)}-\frac{2\cos x}{\cos x(1+\cos x)}dx$

$\Rightarrow$ $\int \sec x-\frac{2}{1+\cos x}dx$

$\Rightarrow$ $\int \sec x-\frac{2}{2{\cos }^2\frac{x}{2}}dx$ since $[\cos x=2{\cos }^2\frac{x}{2}-1]$

$\Rightarrow$ $\int \sec x-{\sec }^2\frac{x}{2}dx$

$\Rightarrow$ $\log (\sec x+\tan x)-2\tan \frac{x}{2}+c$ is our answer.