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

The integral of (e^{2x})/(sqrt(e^x+1)) is 2/3 (e^x+1)^{3/2}-2sqrt(e^x+1)+C

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

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

\frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} - 2 e^{x} + 1 + C

Solution steps

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

Apply u-substitution

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

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

Expand

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

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{2}{3} u^{\frac{3}{2}} - 2 u

Substitute back

u = e^{x} + 1

= \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} - 2 e^{x} + 1

Add a constant to the solution

= \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} - 2 e^{x} + 1 + C

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What is the integral of (e^{2x})/(sqrt(e^x+1)) ?
The integral of (e^{2x})/(sqrt(e^x+1)) is 2/3 (e^x+1)^{3/2}-2sqrt(e^x+1)+C