What is the integral of (x^2)/(x^3-7x+6) ?

The integral of (x^2)/(x^3-7x+6) is -1/4 ln|x-1|+4/5 ln|x-2|+9/20 ln|x+3|+C

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integral of (x^2)/(x^3-7x+6)

\int \frac{x ^{2}}{x ^{3} - 7 x + 6} d x

- \frac{1}{4} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x - 2 ∣ + \frac{9}{20} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{3} - 7 x + 6} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{3} - 7 x + 6} : - \frac{1}{4 ( x - 1 )} + \frac{4}{5 ( x - 2 )} + \frac{9}{20 ( x + 3 )}

= \int - \frac{1}{4 ( x - 1 )} + \frac{4}{5 ( x - 2 )} + \frac{9}{20 ( x + 3 )} 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}{4 ( x - 1 )} d x + \int \frac{4}{5 ( x - 2 )} d x + \int \frac{9}{20 ( x + 3 )} d x

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{4}{5 ( x - 2 )} d x = \frac{4}{5} ln ∣ x - 2 ∣

\int \frac{9}{20 ( x + 3 )} d x = \frac{9}{20} ln ∣ x + 3 ∣

= - \frac{1}{4} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x - 2 ∣ + \frac{9}{20} ln ∣ x + 3 ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ x - 1 ∣ + \frac{4}{5} ln ∣ x - 2 ∣ + \frac{9}{20} ln ∣ x + 3 ∣ + C

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What is the integral of (x^2)/(x^3-7x+6) ?
The integral of (x^2)/(x^3-7x+6) is -1/4 ln|x-1|+4/5 ln|x-2|+9/20 ln|x+3|+C