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

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

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

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

\frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

Solution steps

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

Expand

\frac{1 + x}{3 x} : \frac{1}{3 x} + x^{\frac{2}{3}}

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

Apply the Sum Rule:

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

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

\int \frac{1}{3 x} d x = \frac{3}{2} x^{\frac{2}{3}}

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

= \frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}}

Add a constant to the solution

= \frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

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