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

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

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integral of (x+2/x)^2

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

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

Solution steps

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

Expand

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

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

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

\int 4 d x = 4 x

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

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

Add a constant to the solution

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

\int 1 - \frac{4}{3} x^{\frac{1}{3}} d x

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

\int x^{3} ln (5 x^{2}) d x

\int \frac{5 x - 2}{x ( x - 2 )} d x

\int \frac{4}{( x ^{2} - 16 ) ^{2}} d x

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