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

The integral of (e^{-x})/(1-e^{-x)} is ln|1-e^{-x}|+C

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

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

ln 1 - e^{- x} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 1 - e^{- x}

= ln 1 - e^{- x}

Add a constant to the solution

= ln 1 - e^{- x} + C

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What is the integral of (e^{-x})/(1-e^{-x)} ?
The integral of (e^{-x})/(1-e^{-x)} is ln|1-e^{-x}|+C