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

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

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

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

ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

ln ∣ x - 2 ∣ - \frac{2}{x - 2} - \frac{2}{x - 2} : ln ∣ x - 2 ∣ - \frac{4}{x - 2}

= ln ∣ x - 2 ∣ - \frac{4}{x - 2}

Add a constant to the solution

= ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

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