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

The integral of (x^2+3x+1)/(x^2-2x-3) is 1/4 ln|x+1|+19/4 ln|x-3|+x+C

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

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

\frac{1}{4} ln ∣ x + 1 ∣ + \frac{19}{4} ln ∣ x - 3 ∣ + x + C

Solution steps

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

\frac{x ^{2} + 3 x + 1}{x ^{2} - 2 x - 3} = \frac{5 x + 4}{x ^{2} - 2 x - 3} + 1

= \int \frac{5 x + 4}{x ^{2} - 2 x - 3} + 1 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{5 x + 4}{x ^{2} - 2 x - 3} d x + \int 1 d x

\int \frac{5 x + 4}{x ^{2} - 2 x - 3} d x = \frac{1}{4} ln ∣ x + 1 ∣ + \frac{19}{4} ln ∣ x - 3 ∣

\int 1 d x = x

= \frac{1}{4} ln ∣ x + 1 ∣ + \frac{19}{4} ln ∣ x - 3 ∣ + x

Add a constant to the solution

= \frac{1}{4} ln ∣ x + 1 ∣ + \frac{19}{4} ln ∣ x - 3 ∣ + x + C

What is the integral of (x^2+3x+1)/(x^2-2x-3) ?
The integral of (x^2+3x+1)/(x^2-2x-3) is 1/4 ln|x+1|+19/4 ln|x-3|+x+C