First slide

Methods of integration

Question

$Evaluate \int \frac{e^{x}}{\sqrt{4 - e^{2 x}}} d x$

Easy

A

$\frac{1}{2} \sin^{- 1} (\frac{e^{x}}{2}) + C$
B

$\sin^{- 1} (\frac{e^{x}}{2}) + C$
C

$\sin^{- 1} (\frac{e^{x}}{\sqrt{2}}) + C$
D

None of these

Solution

$\begin{array}{l} I = \int \frac{e^{x}}{\sqrt{4 - e^{2 x}}} d x = \int \frac{e^{x}}{\sqrt{2^{2} - {(e^{x})}^{2}}} d x \\ Let e^{x} = t or e^{x} d x = d t \\ ∴ I = \int \frac{d t}{\sqrt{4 - t^{2}}} = \int \frac{d t}{\sqrt{2^{2} - t^{2}}} \\ = \sin^{- 1} (\frac{t}{2}) + C = \sin^{- 1} (\frac{e^{x}}{2}) + C \end{array}$

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