What is the integral of x(3+2/x)^2 ?

The integral of x(3+2/x)^2 is (9x^2)/2+12x+4ln|x|+C

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integral of x(3+2/x)^2

\int x (3 + \frac{2}{x})^{2} d x

\frac{9 x ^{2}}{2} + 12 x + 4 ln ∣ x ∣ + C

Solution steps

\int x (3 + \frac{2}{x})^{2} d x

Simplify

x (3 + \frac{2}{x})^{2} : 9 x + 12 + \frac{4}{x}

= \int 9 x + 12 + \frac{4}{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 9 x d x + \int 12 d x + \int \frac{4}{x} d x

\int 9 x d x = \frac{9 x ^{2}}{2}

\int 12 d x = 12 x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= \frac{9 x ^{2}}{2} + 12 x + 4 ln ∣ x ∣

Add a constant to the solution

= \frac{9 x ^{2}}{2} + 12 x + 4 ln ∣ x ∣ + C

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What is the integral of x(3+2/x)^2 ?
The integral of x(3+2/x)^2 is (9x^2)/2+12x+4ln|x|+C