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

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

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

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

- \frac{4}{x} + 4 ln ∣ x ∣ + x + C

Solution steps

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

Expand

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

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

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 1 d x = x

= - \frac{4}{x} + 4 ln ∣ x ∣ + x

Add a constant to the solution

= - \frac{4}{x} + 4 ln ∣ x ∣ + x + C

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