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

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

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

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

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

Solution steps

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

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

Add a constant to the solution

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

\int_{0}^{2 π} cos (x) sin (x) d x

x \to - 6 lim (11 x + 3)

f (x) = 12 - x^{2}, (3, 3)

f (x) = \frac{18}{x ^{2} + 2}, (1, 6)

x \to 0 + lim (x^{11 x})

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