What is the integral of sqrt(x)(2x+3\sqrt[3]{x}) ?

The integral of sqrt(x)(2x+3\sqrt[3]{x}) is 4/5 x^{5/2}+18/11 x^{11/6}+C

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integral of sqrt(x)(2x+3\sqrt[3]{x})

\frac{4}{5} x^{\frac{5}{2}} + \frac{18}{11} x^{\frac{11}{6}} + C

Solution steps

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

\int 3 x^{\frac{5}{6}} d x = \frac{18}{11} x^{\frac{11}{6}}

= \frac{4}{5} x^{\frac{5}{2}} + \frac{18}{11} x^{\frac{11}{6}}

Add a constant to the solution

= \frac{4}{5} x^{\frac{5}{2}} + \frac{18}{11} x^{\frac{11}{6}} + C

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What is the integral of sqrt(x)(2x+3\sqrt[3]{x}) ?
The integral of sqrt(x)(2x+3\sqrt[3]{x}) is 4/5 x^{5/2}+18/11 x^{11/6}+C