What is the integral of x^2sqrt((1-x^3)^5) ?

The integral of x^2sqrt((1-x^3)^5) is -2/21 (1-x^3)^{7/2}+C

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integral of x^2sqrt((1-x^3)^5)

\int x^{2} (1 - x^{3})^{5} d x

- \frac{2}{21} (1 - x^{3})^{\frac{7}{2}} + C

Solution steps

\int x^{2} (1 - x^{3})^{5} d x

Apply u-substitution

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

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

Apply the Power Rule

: \frac{2}{7} u^{\frac{7}{2}}

= - \frac{1}{3} \cdot \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = 1 - x^{3}

= - \frac{1}{3} \cdot \frac{2}{7} (1 - x^{3})^{\frac{7}{2}}

Simplify

- \frac{1}{3} \cdot \frac{2}{7} (1 - x^{3})^{\frac{7}{2}} : - \frac{2}{21} (1 - x^{3})^{\frac{7}{2}}

= - \frac{2}{21} (1 - x^{3})^{\frac{7}{2}}

Add a constant to the solution

= - \frac{2}{21} (1 - x^{3})^{\frac{7}{2}} + C

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\int_{- π}^{π} x^{3} sin (n x) d x

7 - x - x^{2}

What is the integral of x^2sqrt((1-x^3)^5) ?
The integral of x^2sqrt((1-x^3)^5) is -2/21 (1-x^3)^{7/2}+C