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

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

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

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

Solution steps

Apply u-substitution

Take the constant out:

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

Expand

Apply the Sum Rule:

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

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

= \frac{1}{9} (\frac{3}{5} v^{\frac{5}{3}} + 3 v^{\frac{2}{3}})

Substitute back

= \frac{1}{9} (\frac{3}{5} (3 (x + 1) - 2)^{\frac{5}{3}} + 3 (3 (x + 1) - 2)^{\frac{2}{3}})

Simplify

\frac{1}{9} (\frac{3}{5} (3 (x + 1) - 2)^{\frac{5}{3}} + 3 (3 (x + 1) - 2)^{\frac{2}{3}}) : \frac{1}{5} (x + 2) (3 x + 1)^{\frac{2}{3}}

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

Add a constant to the solution

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

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