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

The integral of 1/(\sqrt[3]{sqrt(x)+1)} is 6/5 (sqrt(x)+1)^{5/3}-3(sqrt(x)+1)^{2/3}+C

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

\frac{6}{5} (x + 1)^{\frac{5}{3}} - 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

Apply u-substitution

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}}

= 2 (\frac{3}{5} v^{\frac{5}{3}} - \frac{3}{2} v^{\frac{2}{3}})

Substitute back

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

Simplify

2 (\frac{3}{5} (x + 1)^{\frac{5}{3}} - \frac{3}{2} (x + 1)^{\frac{2}{3}}) : \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}}

= \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}}

Add a constant to the solution

= \frac{6}{5} (x + 1)^{\frac{5}{3}} - 3 (x + 1)^{\frac{2}{3}} + C

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What is the integral of 1/(\sqrt[3]{sqrt(x)+1)} ?
The integral of 1/(\sqrt[3]{sqrt(x)+1)} is 6/5 (sqrt(x)+1)^{5/3}-3(sqrt(x)+1)^{2/3}+C