What is the integral of (-2(x-9))/((x+5)(x-2)) ?

The integral of (-2(x-9))/((x+5)(x-2)) is -2(2ln|x+5|-ln|x-2|)+C

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integral of (-2(x-9))/((x+5)(x-2))

\int \frac{- 2 ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

- 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{- 2 ( x - 9 )}{( x + 5 ) ( x - 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 - 9}{( x + 5 ) ( x - 2 )} d x

Take the partial fraction of

\frac{x - 9}{( x + 5 ) ( x - 2 )} : \frac{2}{x + 5} - \frac{1}{x - 2}

= - 2 \cdot \int \frac{2}{x + 5} - \frac{1}{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

= - 2 (\int \frac{2}{x + 5} d x - \int \frac{1}{x - 2} d x)

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

= - 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣)

Add a constant to the solution

= - 2 (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣) + C

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What is the integral of (-2(x-9))/((x+5)(x-2)) ?
The integral of (-2(x-9))/((x+5)(x-2)) is -2(2ln|x+5|-ln|x-2|)+C