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

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

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

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

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

Solution steps

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

Expand

\frac{2 x ^{2} + 3 x}{x ^{2} + 3 x + 2} : \frac{2 x ^{2}}{x ^{2} + 3 x + 2} + \frac{3 x}{x ^{2} + 3 x + 2}

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

\int \frac{2 x ^{2}}{x ^{2} + 3 x + 2} d x = 2 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x)

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

= 2 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x) + 3 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣)

Simplify

2 (ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x) + 3 (- ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣) : 2 x - ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣

= 2 x - ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣

Add a constant to the solution

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

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