What is the integral of (x^2)/(\sqrt[4]{4+x^3)} ?

The integral of (x^2)/(\sqrt[4]{4+x^3)} is 4/9 (4+x^3)^{3/4}+C

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integral of (x^2)/(\sqrt[4]{4+x^3)}

\int \frac{x ^{2}}{4 4 + x ^{3}} d x

\frac{4}{9} (4 + x^{3})^{\frac{3}{4}} + C

Solution steps

\int \frac{x ^{2}}{4 4 + x ^{3}} d x

Apply u-substitution

: \int \frac{1}{3 4 u} d u

= \int \frac{1}{3 4 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}{4 u} d u

Apply the Power Rule

: \frac{4}{3} u^{\frac{3}{4}}

= \frac{1}{3} \cdot \frac{4}{3} u^{\frac{3}{4}}

Substitute back

u = 4 + x^{3}

= \frac{1}{3} \cdot \frac{4}{3} (4 + x^{3})^{\frac{3}{4}}

Simplify

\frac{1}{3} \cdot \frac{4}{3} (4 + x^{3})^{\frac{3}{4}} : \frac{4}{9} (4 + x^{3})^{\frac{3}{4}}

= \frac{4}{9} (4 + x^{3})^{\frac{3}{4}}

Add a constant to the solution

= \frac{4}{9} (4 + x^{3})^{\frac{3}{4}} + C

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What is the integral of (x^2)/(\sqrt[4]{4+x^3)} ?
The integral of (x^2)/(\sqrt[4]{4+x^3)} is 4/9 (4+x^3)^{3/4}+C