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

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

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

\int \frac{e ^{x}}{3 1 + 2 e ^{x}} d x

\frac{3}{4} (1 + 2 e^{x})^{\frac{2}{3}} + C

Solution steps

\int \frac{e ^{x}}{3 1 + 2 e ^{x}} d x

Apply u-substitution

: \int \frac{1}{2 3 u} d u

= \int \frac{1}{2 3 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{3 u} d u

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = 1 + 2 e^{x}

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

Simplify

\frac{1}{2} \cdot \frac{3}{2} (1 + 2 e^{x})^{\frac{2}{3}} : \frac{3}{4} (1 + 2 e^{x})^{\frac{2}{3}}

= \frac{3}{4} (1 + 2 e^{x})^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{4} (1 + 2 e^{x})^{\frac{2}{3}} + C

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