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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int \frac{u + 2}{u ^{2}} d u

= 2 \cdot \int \frac{u + 2}{u ^{2}} d u

Expand

\frac{u + 2}{u ^{2}} : \frac{1}{u} + \frac{2}{u ^{2}}

= 2 \cdot \int \frac{1}{u} + \frac{2}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{1}{u} d u + \int \frac{2}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 2 (ln ∣ u ∣ - \frac{2}{u})

Substitute back

u = x - 2

= 2 (ln ∣ x - 2 ∣ - \frac{2}{x - 2})

Add a constant to the solution

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

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