Question

$\underset{0}{\overset{\pi /4}{\int }}\frac{e^{{\sec }^{2}x}\sin x}{{\cos }^{3}x}dx$ equals

• A. $\frac{e^2}{2}$
• B. $\frac{e^2-1}{2}$
• C. $\frac{e^2+1}{2}$
• D. $\frac{e^2-e}{2}$

Answer 1

The correct option is D $\frac{e^2-e}{2}$

$I=\underset{0}{\overset{\pi /4}{\int }}\frac{e^{{\sec }^{2}x}\sin x}{{\cos }^{3}x}dx$
$=\underset{0}{\overset{\pi /4}{\int }}{e}^{{\sec }^{2}x}{\sec }^{2}x\tan xdx$
Put ${\sec }^{2}x=t$
$\Rightarrow 2{\sec }^{2}x\tan xdx=dt$
$I=\underset{1}{\overset{2}{\int }}{e}^t.\frac{dt}{2}={\left[\frac{e^t}{2}\right]}_1^2$
$\Rightarrow I=\frac{e^2-e}{2}$