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

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

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

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

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

Solution steps

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

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^{t}

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

Add a constant to the solution

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

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