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

The integral of 3x^2\sqrt[4]{9+3x^3} is 4/15 (3x^3+9)^{5/4}+C

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

\frac{4}{15} (3 x^{3} + 9)^{\frac{5}{4}} + C

Solution steps

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply the Power Rule

: \frac{4}{5} u^{\frac{5}{4}}

= 3 \cdot \frac{1}{9} \cdot \frac{4}{5} u^{\frac{5}{4}}

Substitute back

u = 9 + 3 x^{3}

= 3 \cdot \frac{1}{9} \cdot \frac{4}{5} (9 + 3 x^{3})^{\frac{5}{4}}

Simplify

3 \cdot \frac{1}{9} \cdot \frac{4}{5} (9 + 3 x^{3})^{\frac{5}{4}} : \frac{4}{15} (3 x^{3} + 9)^{\frac{5}{4}}

= \frac{4}{15} (3 x^{3} + 9)^{\frac{5}{4}}

Add a constant to the solution

= \frac{4}{15} (3 x^{3} + 9)^{\frac{5}{4}} + C

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What is the integral of 3x^2\sqrt[4]{9+3x^3} ?
The integral of 3x^2\sqrt[4]{9+3x^3} is 4/15 (3x^3+9)^{5/4}+C