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

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

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

\int \frac{e ^{x}}{3 1 + e ^{x}} d x

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

Solution steps

\int \frac{e ^{x}}{3 1 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{3 1 + u} d u

= \int \frac{1}{3 1 + u} d u

Apply u-substitution

: \int \frac{1}{3 v} d v

= \int \frac{1}{3 v} d v

Apply the Power Rule

: \frac{3}{2} v^{\frac{2}{3}}

= \frac{3}{2} v^{\frac{2}{3}}

Substitute back

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

Add a constant to the solution

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

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