Question
Find the value of $\int \frac{e^x d x}{\sqrt{1-e^{2 x}}}$ :
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Answer
Let $\quad I =\int \frac{e^x d x}{\sqrt{1-\left(e^x\right)^2}}$
Let $\quad e^x=t$
$\Rightarrow \quad e^x d x=d t$
$I =\int \frac{d t}{\sqrt{1-t^2}}=\sin ^{-1} t+ C =\sin ^{-1} e^x+ C$
Let $\quad e^x=t$
$\Rightarrow \quad e^x d x=d t$
$I =\int \frac{d t}{\sqrt{1-t^2}}=\sin ^{-1} t+ C =\sin ^{-1} e^x+ C$
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