Question

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

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

Answer 1

Answer: (C)

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