Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration3 Marks
Question
Evaluate : $\int \frac{e^{\frac{x}{2}}}{\sqrt{e^{-x}-e^x}} \cdot d x$
✓
Answer
$ \begin{aligned} & \mathrm{I}=\int \frac{\sqrt{e^x}}{\sqrt{\frac{1}{e^x}-e^x}} \cdot d x \\ & =\int \frac{\sqrt{e^x}}{\sqrt{\frac{1-\left(e^x\right)^2}{e^x}}} \cdot d x \\ & =\int \frac{\sqrt{e^x}}{\frac{\sqrt{1-\left(e^x\right)^2}}{\sqrt{e^x}}} \cdot d x \\ & =\int \frac{\sqrt{e^x} \cdot \sqrt{e^x}}{\sqrt{1-\left(e^x\right)^2}} \cdot d x \\ & =\int \frac{e^x}{\sqrt{1-\left(e^x\right)^2}} \cdot d x \\ & \text { put } e^x=t \\ & \therefore \quad e^x \cdot d x=1 \cdot d t \\ & \mathrm{I}=\int \frac{1}{\sqrt{1-t^2}} \cdot d t \\ & =\sin ^{-1}(t)+c \\ & =\sin ^{-1}\left(e^x\right)+c \\ & \therefore \int \frac{e^{\frac{x}{2}}}{\sqrt{e^{-x}-e^x}} \cdot d x=\sin ^{-1}\left(e^x\right)+c \\ \end{aligned} $
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