Question

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

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

Answer 1

Answer: (C)

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