Question

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

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

Answer 1

Answer: (C)

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