Question

Evaluate $\int ({\text{cosec}}^{2}x\cdot \ln |\sec x|)dx$

(where $C$ is constant of integration)

• A. $x-\cot x\ln |\sec x|+C$
• B. $x+\cot x\ln |\sec x|+C$
• C. $-x-\cot x\ln |\sec x|+C$
• D. $-x+\cot x\ln |\sec x|+C$

Answer 1

The correct option is A $x-\cot x\ln |\sec x|+C$

Let $I=\int ({\text{cosec}}^{2}x\cdot \ln |\sec x|)dx$
Using ILATE rule, we get
$\Rightarrow I=\ln |\sec x|(-\cot x)-\int (-\cot x)\cdot \frac{\sec x\tan x}{\sec x}dx$

$\Rightarrow I=-\cot x\ln |\sec x|+\int dx$

$\therefore I=x-\cot x\ln |\sec x|+C$