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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

Add a constant to the solution

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

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