What is the integral of (x^2)/(x^3+2x^2-11x-12) ?

The integral of (x^2)/(x^3+2x^2-11x-12) is -1/12 ln|x+1|+9/28 ln|x-3|+16/21 ln|x+4|+C

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integral of (x^2)/(x^3+2x^2-11x-12)

\int \frac{x ^{2}}{x ^{3} + 2 x ^{2} - 11 x - 12} d x

- \frac{1}{12} ln ∣ x + 1 ∣ + \frac{9}{28} ln ∣ x - 3 ∣ + \frac{16}{21} ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{3} + 2 x ^{2} - 11 x - 12} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{3} + 2 x ^{2} - 11 x - 12} : - \frac{1}{12 ( x + 1 )} + \frac{9}{28 ( x - 3 )} + \frac{16}{21 ( x + 4 )}

= \int - \frac{1}{12 ( x + 1 )} + \frac{9}{28 ( x - 3 )} + \frac{16}{21 ( x + 4 )} 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{1}{12 ( x + 1 )} d x + \int \frac{9}{28 ( x - 3 )} d x + \int \frac{16}{21 ( x + 4 )} d x

\int \frac{1}{12 ( x + 1 )} d x = \frac{1}{12} ln ∣ x + 1 ∣

\int \frac{9}{28 ( x - 3 )} d x = \frac{9}{28} ln ∣ x - 3 ∣

\int \frac{16}{21 ( x + 4 )} d x = \frac{16}{21} ln ∣ x + 4 ∣

= - \frac{1}{12} ln ∣ x + 1 ∣ + \frac{9}{28} ln ∣ x - 3 ∣ + \frac{16}{21} ln ∣ x + 4 ∣

Add a constant to the solution

= - \frac{1}{12} ln ∣ x + 1 ∣ + \frac{9}{28} ln ∣ x - 3 ∣ + \frac{16}{21} ln ∣ x + 4 ∣ + C

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\int \frac{x}{3 x - 12} d x

What is the integral of (x^2)/(x^3+2x^2-11x-12) ?
The integral of (x^2)/(x^3+2x^2-11x-12) is -1/12 ln|x+1|+9/28 ln|x-3|+16/21 ln|x+4|+C