What is the integral of sqrt(x)(x^3-1/(2x)) ?

The integral of sqrt(x)(x^3-1/(2x)) is -sqrt(x)+2/9 x^{9/2}+C

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integral of sqrt(x)(x^3-1/(2x))

\int x (x^{3} - \frac{1}{2 x}) d x

- x + \frac{2}{9} x^{\frac{9}{2}} + C

Solution steps

\int x (x^{3} - \frac{1}{2 x}) d x

Expand

x (x^{3} - \frac{1}{2 x}) : - \frac{1}{2 x} + x^{\frac{7}{2}}

= \int - \frac{1}{2 x} + x^{\frac{7}{2}} 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}{2 x} d x + \int x^{\frac{7}{2}} d x

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

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

= - x + \frac{2}{9} x^{\frac{9}{2}}

Add a constant to the solution

= - x + \frac{2}{9} x^{\frac{9}{2}} + C

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What is the integral of sqrt(x)(x^3-1/(2x)) ?
The integral of sqrt(x)(x^3-1/(2x)) is -sqrt(x)+2/9 x^{9/2}+C