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

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

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integral of (2x+4)/(x^2(x-2))

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

- 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

= - 2 ln ∣ x ∣ - (- \frac{2}{x}) + 2 ln ∣ x - 2 ∣

Simplify

= - 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= - 2 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 2 ∣ + C

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