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 $$k$$.


ANSWER

$$2$$


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
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