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

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

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

\int \frac{3 + 2 x ^{2}}{4 9 x + 2 x ^{3}} d x

\frac{4}{9} (9 x + 2 x^{3})^{\frac{3}{4}} + C

Solution steps

\int \frac{3 + 2 x ^{2}}{4 9 x + 2 x ^{3}} d x

Apply u-substitution

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

Substitute back

u = 9 x + 2 x^{3}

= \frac{1}{3} \cdot \frac{4}{3} (9 x + 2 x^{3})^{\frac{3}{4}}

Simplify

\frac{1}{3} \cdot \frac{4}{3} (9 x + 2 x^{3})^{\frac{3}{4}} : \frac{4}{9} (9 x + 2 x^{3})^{\frac{3}{4}}

= \frac{4}{9} (9 x + 2 x^{3})^{\frac{3}{4}}

Add a constant to the solution

= \frac{4}{9} (9 x + 2 x^{3})^{\frac{3}{4}} + C

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