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

The integral of (2/x-x)^2 is -4/x-4x+(x^3)/3+C

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integral of (2/x-x)^2

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

- \frac{4}{x} - 4 x + \frac{x ^{3}}{3} + C

Solution steps

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

Simplify

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

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

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

\int 4 d x = 4 x

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

= - \frac{4}{x} - 4 x + \frac{x ^{3}}{3}

Add a constant to the solution

= - \frac{4}{x} - 4 x + \frac{x ^{3}}{3} + C

What is the integral of (2/x-x)^2 ?
The integral of (2/x-x)^2 is -4/x-4x+(x^3)/3+C