What is the integral of (4x^2)/(\sqrt[3]{5-5x)} ?

The integral of (4x^2)/(\sqrt[3]{5-5x)} is -3/(10\sqrt[3]{5)}(5x^2+6x+9)(-x+1)^{2/3}+C

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integral of (4x^2)/(\sqrt[3]{5-5x)}

Solution steps

Take the constant out:

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

Apply u-substitution

Take the constant out:

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

Expand

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 = \frac{3}{8} u^{\frac{8}{3}}

\int 10 u^{\frac{2}{3}} d u = 6 u^{\frac{5}{3}}

= 4 (- \frac{1}{125} (\frac{3}{8} u^{\frac{8}{3}} - 6 u^{\frac{5}{3}} + \frac{75}{2} u^{\frac{2}{3}}))

Substitute back

u = 5 - 5 x

= 4 (- \frac{1}{125} (\frac{3}{8} (5 - 5 x)^{\frac{8}{3}} - 6 (5 - 5 x)^{\frac{5}{3}} + \frac{75}{2} (5 - 5 x)^{\frac{2}{3}}))

Simplify

Add a constant to the solution

What is the integral of (4x^2)/(\sqrt[3]{5-5x)} ?
The integral of (4x^2)/(\sqrt[3]{5-5x)} is -3/(10\sqrt[3]{5)}(5x^2+6x+9)(-x+1)^{2/3}+C