Question

Integral of $\sqrt{1+2cotx(cotx+cosecx)}$ with respect to x is:

• A. 2 In $cos\frac{x}{2}+c$
• B. 2 In $sin\frac{x}{2}+c$
• C. $\frac{1}{2}$ In $cos\frac{x}{2}+c$
• D. In sin x - ln(cosecx+cot x) + c

Answer 1

The correct option is B 2 In $sin\frac{x}{2}+c$

$I=\int \sqrt{1+2cosecxcotx+2co{t}^{2}x}$
$=\int \sqrt{cose{c}^{2}x+2cosecxcotx+co{t}^{2}x}dx=\int (cosecx+cotx)dx$
= Insin x - ln(cosecx+cot x) + c