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

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

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

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

ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{2 x + 5}{( x + 2 ) ( x + 3 )} : \frac{1}{x + 2} + \frac{1}{x + 3}

= \int \frac{1}{x + 2} + \frac{1}{x + 3} 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

= \int \frac{1}{x + 2} d x + \int \frac{1}{x + 3} d x

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

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

= ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣ + C

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