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

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

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

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

1 + e^{t} - ln 1 + e^{t} + C

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{u - 1}{u} : 1 - \frac{1}{u}

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

Apply the Sum Rule:

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = 1 + e^{t}

= 1 + e^{t} - ln 1 + e^{t}

Add a constant to the solution

= 1 + e^{t} - ln 1 + e^{t} + C

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