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

The integral of (1-x^{3/2})^{5/3}*sqrt(x) is -1/4 (1-x^{3/2})^{8/3}+C

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integral of (1-x^{3/2})^{5/3}*sqrt(x)

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

- \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

= - \frac{2}{3} \cdot \frac{3}{8} u^{\frac{8}{3}}

Substitute back

u = 1 - x^{\frac{3}{2}}

= - \frac{2}{3} \cdot \frac{3}{8} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

Simplify

- \frac{2}{3} \cdot \frac{3}{8} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} : - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

= - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}}

Add a constant to the solution

= - \frac{1}{4} (1 - x^{\frac{3}{2}})^{\frac{8}{3}} + C

What is the integral of (1-x^{3/2})^{5/3}*sqrt(x) ?
The integral of (1-x^{3/2})^{5/3}*sqrt(x) is -1/4 (1-x^{3/2})^{8/3}+C