What is the integral of x*\sqrt[3]{5-2x} ?

The integral of x*\sqrt[3]{5-2x} is 1/4 (3/7 (5-2x)^{7/3}-15/4 (5-2x)^{4/3})+C

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integral of x*\sqrt[3]{5-2x}

\frac{1}{4} (\frac{3}{7} (5 - 2 x)^{\frac{7}{3}} - \frac{15}{4} (5 - 2 x)^{\frac{4}{3}}) + C

Solution steps

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{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

= \frac{1}{4} (\frac{3}{7} u^{\frac{7}{3}} - \frac{15}{4} u^{\frac{4}{3}})

Substitute back

u = 5 - 2 x

= \frac{1}{4} (\frac{3}{7} (5 - 2 x)^{\frac{7}{3}} - \frac{15}{4} (5 - 2 x)^{\frac{4}{3}})

Add a constant to the solution

= \frac{1}{4} (\frac{3}{7} (5 - 2 x)^{\frac{7}{3}} - \frac{15}{4} (5 - 2 x)^{\frac{4}{3}}) + C

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What is the integral of x*\sqrt[3]{5-2x} ?
The integral of x*\sqrt[3]{5-2x} is 1/4 (3/7 (5-2x)^{7/3}-15/4 (5-2x)^{4/3})+C