What is the integral of x^2sqrt(5-2x^3) ?

The integral of x^2sqrt(5-2x^3) is -1/9 (5-2x^3)^{3/2}+C

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integral of x^2sqrt(5-2x^3)

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

- \frac{1}{9} (5 - 2 x^{3})^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

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

= \int \frac{5 - 2 u}{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 5 - 2 u d u

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

= \frac{1}{3} (- \frac{1}{2} \cdot \frac{2}{3} (5 - 2 x^{3})^{\frac{3}{2}})

Simplify

\frac{1}{3} (- \frac{1}{2} \cdot \frac{2}{3} (5 - 2 x^{3})^{\frac{3}{2}}) : - \frac{1}{9} (5 - 2 x^{3})^{\frac{3}{2}}

= - \frac{1}{9} (5 - 2 x^{3})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{9} (5 - 2 x^{3})^{\frac{3}{2}} + C

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What is the integral of x^2sqrt(5-2x^3) ?
The integral of x^2sqrt(5-2x^3) is -1/9 (5-2x^3)^{3/2}+C