What is the integral of (e^{2x})/(\sqrt[4]{1+e^x)} ?

The integral of (e^{2x})/(\sqrt[4]{1+e^x)} is 4/7 (1+e^x)^{7/4}-4/3 (1+e^x)^{3/4}+C

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integral of (e^{2x})/(\sqrt[4]{1+e^x)}

\frac{4}{7} (1 + e^{x})^{\frac{7}{4}} - \frac{4}{3} (1 + e^{x})^{\frac{3}{4}} + C

Solution steps

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 u^{\frac{3}{4}} d u = \frac{4}{7} u^{\frac{7}{4}}

= \frac{4}{7} u^{\frac{7}{4}} - \frac{4}{3} u^{\frac{3}{4}}

Substitute back

u = 1 + e^{x}

= \frac{4}{7} (1 + e^{x})^{\frac{7}{4}} - \frac{4}{3} (1 + e^{x})^{\frac{3}{4}}

Add a constant to the solution

= \frac{4}{7} (1 + e^{x})^{\frac{7}{4}} - \frac{4}{3} (1 + e^{x})^{\frac{3}{4}} + C

What is the integral of (e^{2x})/(\sqrt[4]{1+e^x)} ?
The integral of (e^{2x})/(\sqrt[4]{1+e^x)} is 4/7 (1+e^x)^{7/4}-4/3 (1+e^x)^{3/4}+C