What is the integral of e^x(e^x-1)^{-4} ?

The integral of e^x(e^x-1)^{-4} is -1/(3(e^x-1)^3)+C

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integral of e^x(e^x-1)^{-4}

\int e^{x} (e^{x} - 1)^{- 4} d x

- \frac{1}{3 ( e ^{x} - 1 ) ^{3}} + C

Solution steps

\int e^{x} (e^{x} - 1)^{- 4} d x

Apply u-substitution

= \int \frac{1}{( u - 1 ) ^{4}} d u

Apply u-substitution

= \int \frac{1}{v ^{4}} d v

Apply the Power Rule

= - \frac{1}{3 v ^{3}}

Substitute back

= - \frac{1}{3 ( e ^{x} - 1 ) ^{3}}

Add a constant to the solution

= - \frac{1}{3 ( e ^{x} - 1 ) ^{3}} + C

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What is the integral of e^x(e^x-1)^{-4} ?
The integral of e^x(e^x-1)^{-4} is -1/(3(e^x-1)^3)+C