What is the integral of x\sqrt[3]{a+x} ?

The integral of x\sqrt[3]{a+x} is 3/7 (a+x)^{7/3}-a 3/4 (a+x)^{4/3}+C

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integral of x\sqrt[3]{a+x}

\int x 3 a + x d x

\frac{3}{7} (a + x)^{\frac{7}{3}} - a \frac{3}{4} (a + x)^{\frac{4}{3}} + C

Solution steps

\int x 3 a + x d x

Apply u-substitution

: \int (u - a) 3 u d u

= \int (u - a) 3 u d u

Expand

(u - a) 3 u : u^{\frac{4}{3}} - a 3 u

= \int u^{\frac{4}{3}} - a 3 u 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{4}{3}} d u - \int a 3 u d u

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

\int a 3 u d u = a \frac{3}{4} u^{\frac{4}{3}}

= \frac{3}{7} u^{\frac{7}{3}} - a \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = a + x

= \frac{3}{7} (a + x)^{\frac{7}{3}} - a \frac{3}{4} (a + x)^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{7} (a + x)^{\frac{7}{3}} - a \frac{3}{4} (a + x)^{\frac{4}{3}} + C

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