What is the integral of x^2(3-x^3)^{2/3} ?

The integral of x^2(3-x^3)^{2/3} is -1/5 (3-x^3)^{5/3}+C

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integral of x^2(3-x^3)^{2/3}

\int x^{2} (3 - x^{3})^{\frac{2}{3}} d x

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

Solution steps

\int x^{2} (3 - x^{3})^{\frac{2}{3}} d x

Apply u-substitution

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

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

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{3} (- \int v^{\frac{2}{3}} d v)

Apply the Power Rule

: \frac{3}{5} v^{\frac{5}{3}}

= \frac{1}{3} (- \frac{3}{5} v^{\frac{5}{3}})

Substitute back

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

Simplify

\frac{1}{3} (- \frac{3}{5} (3 - x^{3})^{\frac{5}{3}}) : - \frac{1}{5} (3 - x^{3})^{\frac{5}{3}}

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

Add a constant to the solution

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

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What is the integral of x^2(3-x^3)^{2/3} ?
The integral of x^2(3-x^3)^{2/3} is -1/5 (3-x^3)^{5/3}+C