Question

$\int\frac{1}{\sin x\, \cos x}$ dx is equal to

• A. $\log |\tan x| + C$
• B. $\log |\sin 2x| + C$
• C. $\log |\sec x| + C$
• D. $\log |\cos x| + C$
• E. $\log |\sin x| + C$

Answer 1

Answer: (A)

Let $ I=\int\left(\frac{\sin ^{2} x+\cos ^{2} x}{\sin x \cos x}\right) d x $
$=\int\left(\frac{\sin ^{2} x}{\sin x \cos x}+\frac{\cos ^{2} x}{\sin x \cos x} d x\right)$
$=\int(\tan x+\cot x) d x $
$= \log \sec x+(-\log \operatorname{cosec} x)+C $
$= \log \left|\frac{\sec x}{\operatorname{cosec} x}\right|+C=\log |\tan x|+C $