What is the integral of ((3x+11))/((x+2)(x+3)) ?

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

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integral of ((3x+11))/((x+2)(x+3))

\int \frac{( 3 x + 11 )}{( x + 2 ) ( x + 3 )} d x

5 ln ∣ x + 2 ∣ - 2 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{3 x + 11}{( x + 2 ) ( x + 3 )} d x

Take the partial fraction of

\frac{3 x + 11}{( x + 2 ) ( x + 3 )} : \frac{5}{x + 2} - \frac{2}{x + 3}

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

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

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

= 5 ln ∣ x + 2 ∣ - 2 ln ∣ x + 3 ∣

Add a constant to the solution

= 5 ln ∣ x + 2 ∣ - 2 ln ∣ x + 3 ∣ + C

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