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

The integral of x^5(2-5x^3)^{2/3} is 1/1000 (5(-5x^3+2)^{8/3}-16(-5x^3+2)^{5/3})+C

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

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

\frac{1}{1000} (5 (- 5 x^{3} + 2)^{\frac{8}{3}} - 16 (- 5 x^{3} + 2)^{\frac{5}{3}}) + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{\frac{2}{3}} ( u - 2 )}{75} d u

= \int \frac{u ^{\frac{2}{3}} ( u - 2 )}{75} d u

Take the constant out:

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

= \frac{1}{75} \cdot \int u^{\frac{2}{3}} (u - 2) d u

Expand

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

= \frac{1}{75} \cdot \int u^{\frac{5}{3}} - 2 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

= \frac{1}{75} (\int u^{\frac{5}{3}} d u - \int 2 u^{\frac{2}{3}} d u)

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

\int 2 u^{\frac{2}{3}} d u = \frac{6}{5} u^{\frac{5}{3}}

= \frac{1}{75} (\frac{3}{8} u^{\frac{8}{3}} - \frac{6}{5} u^{\frac{5}{3}})

Substitute back

u = 2 - 5 x^{3}

= \frac{1}{75} (\frac{3}{8} (2 - 5 x^{3})^{\frac{8}{3}} - \frac{6}{5} (2 - 5 x^{3})^{\frac{5}{3}})

Simplify

\frac{1}{75} (\frac{3}{8} (2 - 5 x^{3})^{\frac{8}{3}} - \frac{6}{5} (2 - 5 x^{3})^{\frac{5}{3}}) : \frac{1}{1000} (5 (- 5 x^{3} + 2)^{\frac{8}{3}} - 16 (- 5 x^{3} + 2)^{\frac{5}{3}})

= \frac{1}{1000} (5 (- 5 x^{3} + 2)^{\frac{8}{3}} - 16 (- 5 x^{3} + 2)^{\frac{5}{3}})

Add a constant to the solution

= \frac{1}{1000} (5 (- 5 x^{3} + 2)^{\frac{8}{3}} - 16 (- 5 x^{3} + 2)^{\frac{5}{3}}) + C

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What is the integral of x^5(2-5x^3)^{2/3} ?
The integral of x^5(2-5x^3)^{2/3} is 1/1000 (5(-5x^3+2)^{8/3}-16(-5x^3+2)^{5/3})+C