What is the integral of (3x^2+19x+27)/((x+2)(x+3)^2) ?

The integral of (3x^2+19x+27)/((x+2)(x+3)^2) is ln|x+2|+2ln|x+3|-3/(x+3)+C

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integral of (3x^2+19x+27)/((x+2)(x+3)^2)

\int \frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} d x

ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3} + C

Solution steps

\int \frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} d x

Take the partial fraction of

\frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} : \frac{1}{x + 2} + \frac{2}{x + 3} + \frac{3}{( x + 3 ) ^{2}}

= \int \frac{1}{x + 2} + \frac{2}{x + 3} + \frac{3}{( x + 3 ) ^{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

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

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

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

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

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3}

Add a constant to the solution

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ - \frac{3}{x + 3} + C

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What is the integral of (3x^2+19x+27)/((x+2)(x+3)^2) ?
The integral of (3x^2+19x+27)/((x+2)(x+3)^2) is ln|x+2|+2ln|x+3|-3/(x+3)+C