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

The integral of sqrt(x)(3x-2/x) is -4sqrt(x)+6/5 x^{5/2}+C

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

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

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

Solution steps

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

Expand

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

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

\int \frac{2}{x} d x = 4 x

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

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

Add a constant to the solution

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

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\int \frac{4 x}{( 2 x ^{2} + 5 ) ^{4}} d x

\int \frac{x ^{2}}{5 - x ^{6}} d x

\int \frac{1 + tan ( x )}{1 - cos ( x )} d x

What is the integral of sqrt(x)(3x-2/x) ?
The integral of sqrt(x)(3x-2/x) is -4sqrt(x)+6/5 x^{5/2}+C