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

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

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

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

- 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{2}{( x + 2 ) ( x - 3 )} d x = 2 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

\int \frac{4 x}{( x + 2 ) ( x - 3 )} d x = 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣)

= 2 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣)

Simplify

2 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣) : - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣

= - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣

Add a constant to the solution

= - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣ + C

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