What is the integral of (1+\sqrt[3]{x})/(sqrt(x)) ?

The integral of (1+\sqrt[3]{x})/(sqrt(x)) is 2sqrt(x)+6/5 x^{5/6}+C

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integral of (1+\sqrt[3]{x})/(sqrt(x))

\int \frac{1 + 3 x}{x} d x

2 x + \frac{6}{5} x^{\frac{5}{6}} + C

Solution steps

\int \frac{1 + 3 x}{x} d x

Expand

\frac{1 + 3 x}{x} : \frac{1}{x} + \frac{1}{6 x}

= \int \frac{1}{x} + \frac{1}{6 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 \frac{1}{x} d x + \int \frac{1}{6 x} d x

\int \frac{1}{x} d x = 2 x

\int \frac{1}{6 x} d x = \frac{6}{5} x^{\frac{5}{6}}

= 2 x + \frac{6}{5} x^{\frac{5}{6}}

Add a constant to the solution

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

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What is the integral of (1+\sqrt[3]{x})/(sqrt(x)) ?
The integral of (1+\sqrt[3]{x})/(sqrt(x)) is 2sqrt(x)+6/5 x^{5/6}+C