What is the integral of ((-4x-2)/((x-3)^2)) ?

The integral of ((-4x-2)/((x-3)^2)) is -4ln|x-3|+(14)/(x-3)+C

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integral of ((-4x-2)/((x-3)^2))

\int (\frac{- 4 x - 2}{( x - 3 ) ^{2}}) d x

- 4 ln ∣ x - 3 ∣ + \frac{14}{x - 3} + C

Solution steps

\int \frac{- 4 x - 2}{( x - 3 ) ^{2}} d x

Expand

\frac{- 4 x - 2}{( x - 3 ) ^{2}} : - \frac{4 x}{( x - 3 ) ^{2}} - \frac{2}{( x - 3 ) ^{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{4 x}{( x - 3 ) ^{2}} d x - \int \frac{2}{( x - 3 ) ^{2}} d x

\int \frac{4 x}{( x - 3 ) ^{2}} d x = 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3})

\int \frac{2}{( x - 3 ) ^{2}} d x = - \frac{2}{x - 3}

= - 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{2}{x - 3})

Simplify

- 4 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{2}{x - 3}) : - 4 ln ∣ x - 3 ∣ + \frac{14}{x - 3}

= - 4 ln ∣ x - 3 ∣ + \frac{14}{x - 3}

Add a constant to the solution

= - 4 ln ∣ x - 3 ∣ + \frac{14}{x - 3} + C

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What is the integral of ((-4x-2)/((x-3)^2)) ?
The integral of ((-4x-2)/((x-3)^2)) is -4ln|x-3|+(14)/(x-3)+C