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

The integral of x/(\sqrt[3]{3x+1)} is 1/2 x(3x+1)^{2/3}-1/10 (3x+1)^{5/3}+C

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

\frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}} + C

Solution steps

Apply Integration By Parts

: \frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \int \frac{1}{2} (3 x + 1)^{\frac{2}{3}} d x

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

\int \frac{1}{2} (3 x + 1)^{\frac{2}{3}} d x = \frac{1}{10} (3 x + 1)^{\frac{5}{3}}

= \frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}}

Add a constant to the solution

= \frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}} + C

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What is the integral of x/(\sqrt[3]{3x+1)} ?
The integral of x/(\sqrt[3]{3x+1)} is 1/2 x(3x+1)^{2/3}-1/10 (3x+1)^{5/3}+C