Question

The value of $\int e^{\sec x} \cdot \sec ^3 x\left(\sin ^2 x+\cos x+\sin x+\sin x \cos x\right) d x$ is :

• A. $e^{\sec x}\left(\sec ^2 x+\sec x \tan x\right)+c$
• B. $e^{sec x}+c$
• C. $e^{sec x}\left(\right.secx+tanx\left.\right)+c$
• D. None of these

Answer 1

Answer: (C)

$I =\int e ^{\sec x} \sec ^3 x\left(\sin ^2 x+\cos x+\sin x+\sin x \cos x\right) d x$
$\Rightarrow I =\int e ^{\sec x} \cdot \sec x\left(\tan ^2 x+\sec x+\sec x \tan x+\tan x\right) d x$
$=\int e ^{\sec x} \cdot \sec x(\tan x+1)(\sec x+\tan x) d x$
$=\int e ^{\sec x} \sec x \tan x(\sec x+\tan x) d x+\int e ^{\sec x} \cdot\left(\sec ^2 x+\sec x \tan x\right) d x$
$=(\sec x+\tan x) e^{\sec x}-\int e ^{\sec x} \cdot\left(\sec x \tan x+\sec ^2 x\right) d x+\int e ^{\sec x}\left(\sec x \tan x+\sec ^2 x\right)$
$= e ^{\sec x}(\sec x+\tan x)+c$