Question

Evaluate $\int \frac{1+\cos x}{\sin x\cos x}$

• A. $\log \mid \sin x\mid +\log \mid cosx\mid +c$
• B. $\log \mid \tan x.\tan \frac{x}{2}\mid +c$
• C. $\log \mid 1+\tan \frac{x}{2}\mid +c$
• D. $\log \mid \sec \frac{x}{2}+\tan \frac{x}{2}\mid +c$

Answer 1

The correct option is B $\log \mid \tan x.\tan \frac{x}{2}\mid +c$

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

$\int \frac{1}{\sin x\cos x}dx+\int \frac{\cos x}{\sin x\cos x}dx$

$2\int \frac{dx}{2\sin x\cos x}dx+\int \text{cosec}xdx$

$2\int \text{cosec}2xdx+\int \text{cosec}xdx$

$2\frac{\log \mid \text{cosec}2x-\cot 2x\mid }{2}+\log \mid \text{cosec}x-\cot x\mid +c$

$\log \mid (\text{cosec}2x-\cot 2x)+(\text{cosec}x-\cot x)\mid +c$

$\log \mid \left(\frac{1-\cos 2x}{\sin 2x}\right)\left(\frac{1-\cos x}{\sin x}\right)\mid +c$

$\log \mid \frac{2{\sin }^{2}x}{2\sin x\cos x}\frac{2{\sin }^2\frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}\mid +c$

$\log \mid \tan x.\tan \frac{x}{2}\mid +c$