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

The integral of ((2x-sqrt(x))^2)/(sqrt(x)) is 8/5 x^{5/2}-2x^2+2/3 x^{3/2}+C

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

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

\frac{8}{5} x^{\frac{5}{2}} - 2 x^{2} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Factor

2 x - x : - (- 2 x + x)

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

(- (- 2 x + x))^{2} = (- 2 x + x)^{2}

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

Expand

\frac{( - 2 x + x ) ^{2}}{x} : 4 x^{\frac{3}{2}} - 4 x + x

= \int 4 x^{\frac{3}{2}} - 4 x + 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 4 x^{\frac{3}{2}} d x - \int 4 x d x + \int x d x

\int 4 x^{\frac{3}{2}} d x = \frac{8}{5} x^{\frac{5}{2}}

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

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

= \frac{8}{5} x^{\frac{5}{2}} - 2 x^{2} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{8}{5} x^{\frac{5}{2}} - 2 x^{2} + \frac{2}{3} x^{\frac{3}{2}} + C

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What is the integral of ((2x-sqrt(x))^2)/(sqrt(x)) ?
The integral of ((2x-sqrt(x))^2)/(sqrt(x)) is 8/5 x^{5/2}-2x^2+2/3 x^{3/2}+C