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

Solution:

We have, 

$\begin{array}{l}1=\int e^{\sec x}\left({\tan }^{2}x\sec x\right.+{\sec }^{2}x\\ \left. +\tan x{\sec }^{2}x+\tan x\sec x\right)dx\\ \Rightarrow I=\int e^{\sec x}[\sec x\tan x(\sec x+\tan x)\\ \left. +\left(\sec x\tan x+{\sec }^{2}x\right)\right]dx\end{array}$

$\Rightarrow I=\int e^{\sec x}\underset{II}{\sec x\tan x}\cdot \left(\underset{I}{\sec x+\tan x}\right)dx$

                                     $+\int e^{\sec x}\cdot \left(\sec x\tan x+{\sec }^{2}x\right)dx$

$\begin{array}{l}\Rightarrow I=(\sec x+\tan x)e^{\sec x}-\int \left(\sec x\tan x+{\sec }^{2}x\right)e^{\sec x}dx\\ +\int e^{\sec x}\left(\sec x\tan x+{\sec }^{2}x\right)dx\\ \Rightarrow I=e^{\sec x}(\sec x+\tan x)+C\end{array}$