What is the integral of x^2\sqrt[3]{x^3+5} ?

The integral of x^2\sqrt[3]{x^3+5} is 1/4 (x^3+5)^{4/3}+C

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integral of x^2\sqrt[3]{x^3+5}

\int x^{2} 3 x^{3} + 5 d x

\frac{1}{4} (x^{3} + 5)^{\frac{4}{3}} + C

Solution steps

\int x^{2} 3 x^{3} + 5 d x

Apply u-substitution

: \int \frac{u 3 u ^{\frac{3}{2}} + 5}{2 x} d u

= \int \frac{3 u + 5}{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 3 u + 5 d u

Apply u-substitution

: \int 3 v d v

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

Apply the Power Rule

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of x^2\sqrt[3]{x^3+5} ?
The integral of x^2\sqrt[3]{x^3+5} is 1/4 (x^3+5)^{4/3}+C