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

The integral of sqrt(x)(x^2-1/x) is -2sqrt(x)+2/7 x^{7/2}+C

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

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

- 2 x + \frac{2}{7} x^{\frac{7}{2}} + C

Solution steps

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

Expand

x (x^{2} - \frac{1}{x}) : - \frac{1}{x} + x^{\frac{5}{2}}

= \int - \frac{1}{x} + x^{\frac{5}{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}{x} d x + \int x^{\frac{5}{2}} d x

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= - 2 x + \frac{2}{7} x^{\frac{7}{2}}

Add a constant to the solution

= - 2 x + \frac{2}{7} x^{\frac{7}{2}} + C

What is the integral of sqrt(x)(x^2-1/x) ?
The integral of sqrt(x)(x^2-1/x) is -2sqrt(x)+2/7 x^{7/2}+C