What is the integral of u^2sqrt(1-u^3) ?

The integral of u^2sqrt(1-u^3) is -2/9 (1-u^3)^{3/2}+C

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integral of u^2sqrt(1-u^3)

\int u^{2} 1 - u^{3} d u

- \frac{2}{9} (1 - u^{3})^{\frac{3}{2}} + C

Solution steps

\int u^{2} 1 - u^{3} d u

Apply u-substitution

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

= \int - \frac{v}{3} d v

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

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

Substitute back

v = 1 - u^{3}

= - \frac{1}{3} \cdot \frac{2}{3} (1 - u^{3})^{\frac{3}{2}}

Simplify

- \frac{1}{3} \cdot \frac{2}{3} (1 - u^{3})^{\frac{3}{2}} : - \frac{2}{9} (1 - u^{3})^{\frac{3}{2}}

= - \frac{2}{9} (1 - u^{3})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{9} (1 - u^{3})^{\frac{3}{2}} + C

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What is the integral of u^2sqrt(1-u^3) ?
The integral of u^2sqrt(1-u^3) is -2/9 (1-u^3)^{3/2}+C