What is the integral of (x^2+7)/(x^3-3x^2-x+3) ?

The integral of (x^2+7)/(x^3-3x^2-x+3) is 2ln|x-3|+ln|x+1|-2ln|x-1|+C

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integral of (x^2+7)/(x^3-3x^2-x+3)

\int \frac{x ^{2} + 7}{x ^{3} - 3 x ^{2} - x + 3} d x

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

Solution steps

\int \frac{x ^{2} + 7}{x ^{3} - 3 x ^{2} - x + 3} d x

Take the partial fraction of

\frac{x ^{2} + 7}{x ^{3} - 3 x ^{2} - x + 3} : \frac{2}{x - 3} + \frac{1}{x + 1} - \frac{2}{x - 1}

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

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

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

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

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

Add a constant to the solution

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

What is the integral of (x^2+7)/(x^3-3x^2-x+3) ?
The integral of (x^2+7)/(x^3-3x^2-x+3) is 2ln|x-3|+ln|x+1|-2ln|x-1|+C