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

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

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

Solution steps

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{1}{v ^{\frac{2}{3}}} d v

= \frac{1}{3} \cdot \int \frac{1}{v ^{\frac{2}{3}}} d v

Apply the Power Rule

: 3 v^{\frac{1}{3}}

= \frac{1}{3} \cdot 3 v^{\frac{1}{3}}

Substitute back

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

Simplify

Add a constant to the solution

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