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

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

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

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

\frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}} + C

Solution steps

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

Apply u-substitution

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

= \int - (- u + 1) u^{\frac{2}{3}} d u

Take the constant out:

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

= - \int (- u + 1) u^{\frac{2}{3}} d u

Expand

(- u + 1) u^{\frac{2}{3}} : - u^{\frac{5}{3}} + u^{\frac{2}{3}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

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

= - (- \frac{3}{8} u^{\frac{8}{3}} + \frac{3}{5} u^{\frac{5}{3}})

Substitute back

u = 1 - x

= - (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}})

Simplify

= \frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}}

Add a constant to the solution

= \frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}} + C

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