What is the integral of ((2x-5))/((x+2)(x+3)) ?

The integral of ((2x-5))/((x+2)(x+3)) is -9ln|x+2|+11ln|x+3|+C

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integral of ((2x-5))/((x+2)(x+3))

\int \frac{( 2 x - 5 )}{( x + 2 ) ( x + 3 )} d x

- 9 ln ∣ x + 2 ∣ + 11 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{2 x - 5}{( x + 2 ) ( x + 3 )} d x

Expand

\frac{2 x - 5}{( x + 2 ) ( x + 3 )} : \frac{2 x}{( x + 2 ) ( x + 3 )} - \frac{5}{( 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}{( x + 2 ) ( x + 3 )} d x - \int \frac{5}{( x + 2 ) ( x + 3 )} d x

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

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

= 2 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) - 5 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣)

Simplify

2 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) - 5 (ln ∣ x + 2 ∣ - ln ∣ x + 3 ∣) : - 9 ln ∣ x + 2 ∣ + 11 ln ∣ x + 3 ∣

= - 9 ln ∣ x + 2 ∣ + 11 ln ∣ x + 3 ∣

Add a constant to the solution

= - 9 ln ∣ x + 2 ∣ + 11 ln ∣ x + 3 ∣ + C

x \to 0 - lim (x [x] cot (x))

\frac{d}{d x} (\frac{x ^{2} + 1}{x ^{2} - 1})

\int_{0}^{1} \frac{1}{x ^{- 2}} d x

\frac{d}{d x} (x^{2} cos (\frac{1}{x}))

t^{^{'}}

What is the integral of ((2x-5))/((x+2)(x+3)) ?
The integral of ((2x-5))/((x+2)(x+3)) is -9ln|x+2|+11ln|x+3|+C