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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

Add a constant to the solution

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

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