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

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

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integral of (e^x)/(sqrt(e^x-1))

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

2 e^{x} - 1 + C

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

= 2 u^{\frac{1}{2}}

Substitute back

u = e^{x} - 1

= 2 (e^{x} - 1)^{\frac{1}{2}}

Apply radical rule:

= 2 e^{x} - 1

Add a constant to the solution

= 2 e^{x} - 1 + C

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