Question

$\underset{-1}{\overset{1/2}{\int }}\frac{e^x\left(2-x^2\right)dx}{(1-x)\sqrt{1-x^2}}$ is equal to

• A. $\frac{\sqrt{e}}{2}(\sqrt{3}+1)$
• B. $\frac{\sqrt{3e}}{2}$
• C. $\sqrt{3e}$
• D. $\sqrt{\frac{e}{3}}$

Answer 1

Answer: (C)

$\underset{-1}{\overset{1/2}{\int }}\frac{e^x\left(2-x^2\right)dx}{(1-x)\sqrt{1-x^2}}$
$=\underset{-1}{\overset{1/2}{\int }}\frac{e^x\left(1-x^2+1\right)}{(1-x)\sqrt{1-x^2}}$
$=\underset{-1}{\overset{1/2}{\int }}{e}^x\left[\sqrt{\frac{1+x}{1-x}}+\frac{1}{(1-x)\sqrt{1-x^2}}\right]dx$
$={e^x\sqrt{\frac{1+x}{1-x}}|}_{-1}^{1/2}$
$=\sqrt{3e}$