What is the integral of (x^2+1)/(x^2+3x+2) ?

The integral of (x^2+1)/(x^2+3x+2) is ln|x+1|-4ln|x+2|+x-ln|2x+4|+ln|2x+2|+C

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integral of (x^2+1)/(x^2+3x+2)

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

ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x - ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣ + C

Solution steps

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

Expand

\frac{x ^{2} + 1}{x ^{2} + 3 x + 2} : \frac{x ^{2}}{x ^{2} + 3 x + 2} + \frac{1}{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{x ^{2}}{x ^{2} + 3 x + 2} d x + \int \frac{1}{x ^{2} + 3 x + 2} d x

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

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

= ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x - ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣

Add a constant to the solution

= ln ∣ x + 1 ∣ - 4 ln ∣ x + 2 ∣ + x - ln ∣ 2 x + 4 ∣ + ln ∣ 2 x + 2 ∣ + C

What is the integral of (x^2+1)/(x^2+3x+2) ?
The integral of (x^2+1)/(x^2+3x+2) is ln|x+1|-4ln|x+2|+x-ln|2x+4|+ln|2x+2|+C