What is the integral of x\sqrt[3]{x+5} ?

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

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integral of x\sqrt[3]{x+5}

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

Solution steps

Apply u-substitution

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{3}{7} u^{\frac{7}{3}} - \frac{15}{4} u^{\frac{4}{3}}

Substitute back

u = x + 5

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

Add a constant to the solution

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

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