What is the integral of (e^{-x})/(sqrt(1-e^{-2x))} ?

The integral of (e^{-x})/(sqrt(1-e^{-2x))} is arcsin(sqrt(1-e^{-2x)})+C

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integral of (e^{-x})/(sqrt(1-e^{-2x))}

\int \frac{e ^{- x}}{1 - e ^{- 2 x}} d x

arcsin (1 - e^{- 2 x}) + C

Solution steps

\int \frac{e ^{- x}}{1 - e ^{- 2 x}} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 1} d u

= \int \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = arcsin (u)

= arcsin (u)

Substitute back

u = 1 - e^{- 2 x}

= arcsin (1 - e^{- 2 x})

Add a constant to the solution

= arcsin (1 - e^{- 2 x}) + C

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What is the integral of (e^{-x})/(sqrt(1-e^{-2x))} ?
The integral of (e^{-x})/(sqrt(1-e^{-2x))} is arcsin(sqrt(1-e^{-2x)})+C