What is the integral of (e^{2x})/(\sqrt[3]{1+e^x)} ?

The integral of (e^{2x})/(\sqrt[3]{1+e^x)} is 3/5 (1+e^x)^{5/3}-3/2 (1+e^x)^{2/3}+C

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integral of (e^{2x})/(\sqrt[3]{1+e^x)}

\frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

Solution steps

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply u-substitution

Expand

Apply the Sum Rule:

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

\int w^{\frac{2}{3}} d w = \frac{3}{5} w^{\frac{5}{3}}

= \frac{1}{2} \cdot 2 (\frac{3}{5} w^{\frac{5}{3}} - \frac{3}{2} w^{\frac{2}{3}})

Substitute back

= \frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}})

Simplify

\frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}}) : \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

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What is the integral of (e^{2x})/(\sqrt[3]{1+e^x)} ?
The integral of (e^{2x})/(\sqrt[3]{1+e^x)} is 3/5 (1+e^x)^{5/3}-3/2 (1+e^x)^{2/3}+C