What is the integral of (e^x)/(e^x-2) ?

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

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integral of (e^x)/(e^x-2)

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

ln ∣ e^{x} - 2 ∣ + C

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = e^{x} - 2

= ln ∣ e^{x} - 2 ∣

Add a constant to the solution

= ln ∣ e^{x} - 2 ∣ + C

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