What is the integral of (27x^3-2)/(\sqrt[3]{x)} ?

The integral of (27x^3-2)/(\sqrt[3]{x)} is 81/11 x^{11/3}-3x^{2/3}+C

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integral of (27x^3-2)/(\sqrt[3]{x)}

\int \frac{27 x ^{3} - 2}{3 x} d x

\frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}} + C

Solution steps

\int \frac{27 x ^{3} - 2}{3 x} d x

Expand

\frac{27 x ^{3} - 2}{3 x} : 27 x^{\frac{8}{3}} - \frac{2}{3 x}

= \int 27 x^{\frac{8}{3}} - \frac{2}{3 x} 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 27 x^{\frac{8}{3}} d x - \int \frac{2}{3 x} d x

\int 27 x^{\frac{8}{3}} d x = \frac{81}{11} x^{\frac{11}{3}}

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

= \frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}}

Add a constant to the solution

= \frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}} + C

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What is the integral of (27x^3-2)/(\sqrt[3]{x)} ?
The integral of (27x^3-2)/(\sqrt[3]{x)} is 81/11 x^{11/3}-3x^{2/3}+C