What is the integral of ((3x^2+6x+2))/(x^2+3x+2) ?

The integral of ((3x^2+6x+2))/(x^2+3x+2) is 3x-ln|x+1|-2ln|x+2|+C

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integral of ((3x^2+6x+2))/(x^2+3x+2)

\int \frac{( 3 x ^{2} + 6 x + 2 )}{x ^{2} + 3 x + 2} d x

3 x - ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{3 x ^{2} + 6 x + 2}{x ^{2} + 3 x + 2} d x

Take the partial fraction of

\frac{3 x ^{2} + 6 x + 2}{x ^{2} + 3 x + 2} : 3 - \frac{1}{x + 1} - \frac{2}{x + 2}

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

\int 3 d x = 3 x

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

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

= 3 x - ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣

Add a constant to the solution

= 3 x - ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣ + C

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What is the integral of ((3x^2+6x+2))/(x^2+3x+2) ?
The integral of ((3x^2+6x+2))/(x^2+3x+2) is 3x-ln|x+1|-2ln|x+2|+C