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

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

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integral of x^2(2-x^3)^{1/2}

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u ( 2 - u ^{\frac{3}{2}} ) ^{\frac{1}{2}}}{2 x} d u

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

Apply u-substitution

: \int - v^{\frac{1}{2}} d v

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

Apply the Power Rule

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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