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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

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

Substitute back

u = x^{3} + 1

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

Simplify

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

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

Add a constant to the solution

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

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