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

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

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

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

2 ln ∣ e^{x} - 1 ∣ - x + C

Solution steps

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

\frac{e ^{x} + 1}{e ^{x} - 1} = \frac{2 e ^{x}}{e ^{x} - 1} - 1

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d x = x

= 2 ln ∣ e^{x} - 1 ∣ - x

Add a constant to the solution

= 2 ln ∣ e^{x} - 1 ∣ - x + C

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