What is the integral of xsqrt((8-5x^2)^3) ?

The integral of xsqrt((8-5x^2)^3) is -1/25 (8-5x^2)^{5/2}+C

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integral of xsqrt((8-5x^2)^3)

\int x (8 - 5 x^{2})^{3} d x

- \frac{1}{25} (8 - 5 x^{2})^{\frac{5}{2}} + C

Solution steps

\int x (8 - 5 x^{2})^{3} d x

Apply u-substitution

: \int - \frac{u ^{\frac{3}{2}}}{10} d u

= \int - \frac{u ^{\frac{3}{2}}}{10} d u

Take the constant out:

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

= - \frac{1}{10} \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= - \frac{1}{10} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 8 - 5 x^{2}

= - \frac{1}{10} \cdot \frac{2}{5} (8 - 5 x^{2})^{\frac{5}{2}}

Simplify

- \frac{1}{10} \cdot \frac{2}{5} (8 - 5 x^{2})^{\frac{5}{2}} : - \frac{1}{25} (8 - 5 x^{2})^{\frac{5}{2}}

= - \frac{1}{25} (8 - 5 x^{2})^{\frac{5}{2}}

Add a constant to the solution

= - \frac{1}{25} (8 - 5 x^{2})^{\frac{5}{2}} + C

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What is the integral of xsqrt((8-5x^2)^3) ?
The integral of xsqrt((8-5x^2)^3) is -1/25 (8-5x^2)^{5/2}+C