What is the integral of (e^x)/(1-e^{2x)} ?

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

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integral of (e^x)/(1-e^{2x)}

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

\frac{ln ∣ e ^{x} + 1 ∣}{2} - \frac{ln ∣ e ^{x} - 1 ∣}{2} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{1 - u ^{2}} d u

= \int \frac{1}{1 - u ^{2}} d u

Use the common integral:

\int \frac{1}{1 - u ^{2}} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

Substitute back

u = e^{x}

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

Add a constant to the solution

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

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