Definite IntegrationHard
Question
Let $f(x) = \int_{}^{}\ \frac{\left( 2 - x^{2} \right) \cdot e^{x}}{(\sqrt{1 + x})(1 - x)^{3/2}}dx$. If $f(0) - 0$, then $f\left( \frac{1}{2} \right)$ is equal to :
Options
A.$\sqrt{3e} - 1$
B.$\sqrt{2e} + 1$
C.$\sqrt{2e} - 1$
D.$\sqrt{3e} + 1$
Solution
$\int_{}^{}\ e^{x}\left( \frac{\left( 1 - x^{2} \right) + 1}{\sqrt{1 + x} \cdot (1 - x)^{3/2}} \right)dx$
$${\int e^{x}\left( \frac{\left( 1 - x^{2} \right)}{\sqrt{1 + x} \cdot (1 - x)^{3/2}} + \frac{1}{\sqrt{1 + x} \cdot (1 - x)^{3/2}} \right)dx }{\int e^{x}\left( \sqrt{\frac{1 + x}{1 - x}} + \frac{1}{\sqrt{1 + x} \cdot (1 - x)^{3/2}} \right)dx }{= e^{x}\sqrt{\frac{1 + x}{1 - x}} + C }{f(x) = e^{x}\sqrt{\frac{1 + x}{1 - x}} - 1 }{f\left( \frac{1}{2} \right) = \sqrt{3e} - 1}$$
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