Question

$\displaystyle \int e^{sec x}\cdot \left(sec\right)^{3}x\left(\left(sin\right)^{2} x + cos x + sin x + sin x cos x\right)dx$ is equal to (where $c$ is the constant of integration)

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

Answer 1

Answer: (C)

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