1 x ^2 x d x= - Vidyadip

MCQ

$\int \frac{1}{\sin x \cdot \cos ^2 x} d x=$

A
$\sec x \cdot \tan x+c$
B
$\sec x+\log |\sec x+\tan x|+c$
C
$\sec x+\log |\sec x-\tan x|+c$
✓
$\sec x+\log |\operatorname{cosec} x-\cot x|+c$

✓

Answer

Correct option: D.

$\sec x+\log |\operatorname{cosec} x-\cot x|+c$

Let $I=\int \frac{1}{\sin x \cos ^2 x} d x$
$=\int \frac{\cos ^2 x+\sin ^2 x}{\sin x \cos ^2 x} d x$
$=\int \frac{1}{\sin x} d x+\int \frac{\sin x}{\cos ^2 x} d x s$
$=\int \operatorname{cosec} x \ d x+\int \tan x \sec x \ d x$
$=\log |\operatorname{cosec} x-\cot x|+\sec x+c$

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