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

The integral of (2x-1/(sqrt(x)))^2 is ln|x|-8/3 x^{3/2}+(4x^3)/3+C

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

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

ln ∣ x ∣ - \frac{8}{3} x^{\frac{3}{2}} + \frac{4 x ^{3}}{3} + C

Solution steps

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

Expand

(2 x - \frac{1}{x})^{2} : \frac{1}{x} - 4 x^{\frac{1}{2}} + 4 x^{2}

= \int \frac{1}{x} - 4 x^{\frac{1}{2}} + 4 x^{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 4 x^{\frac{1}{2}} d x + \int 4 x^{2} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int 4 x^{\frac{1}{2}} d x = \frac{8}{3} x^{\frac{3}{2}}

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

= ln ∣ x ∣ - \frac{8}{3} x^{\frac{3}{2}} + \frac{4 x ^{3}}{3}

Add a constant to the solution

= ln ∣ x ∣ - \frac{8}{3} x^{\frac{3}{2}} + \frac{4 x ^{3}}{3} + C

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What is the integral of (2x-1/(sqrt(x)))^2 ?
The integral of (2x-1/(sqrt(x)))^2 is ln|x|-8/3 x^{3/2}+(4x^3)/3+C