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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = 2 + 3 x

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

Simplify

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

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

Add a constant to the solution

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

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