Mathematics

# Evaluate $\displaystyle \int \frac{e^{x}}{\sqrt{4-e^{2x}}}dx.$. The solution is$\displaystyle \sin^{-1}\left ( \frac{e^{x}}{k} \right )+C$ Find $k$.

$2$

##### SOLUTION
$\displaystyle I=\int \frac{e^{x}}{\sqrt{4-e^{2x}}}dx=\int \frac{e^{x}}{\sqrt{2^{2}-\left ( e^{x} \right )^{2}}}dx$
Let $\displaystyle e^{x}=t$ or $\displaystyle e^{x}\:dx=dt$
$\displaystyle \therefore I=\int \frac{dt}{\sqrt{4-t^{2}}}=\int \frac{dt}{\sqrt{2^{2}-t^{2}}}$
$\displaystyle =\sin^{-1}\left ( \frac{t}{2} \right )+C=\sin^{-1}\left ( \frac{e^{x}}{2} \right )+C$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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