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Theorems on Integration

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Question

Solve:
$\int \frac{d x}{\sqrt{1 - e^{2 x}}}$

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Solution

$\begin{matrix} \int \frac{d x}{\sqrt{1 - e^{2 x}}} M u l t i p l y i n g t h e n u m e r a t o r a n d d e n o m e r a t o r b y e^{- x}, w e g e t \int \frac{d x}{\sqrt{1 - e^{2 x}}} = \int \frac{e^{- x} d x}{e^{- x} {(1 - e^{2 x})}^{\frac{1}{2}}} = - \int \frac{e^{- x}}{\sqrt{e^{- 2 x} - 1}} d x = - \int \frac{d t}{\sqrt{t^{2} - 1}} [p u t t i n g e^{- x} = t] = log ∣ ∣ t + \sqrt{t^{2} - 1} ∣ ∣ + C = - log ∣ ∣ e^{- x} + \sqrt{e^{- 2 x} - 1} ∣ ∣ + C \end{matrix}$

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