What is the integral of (x^2-19)/(x^3-2x^2+x) ?

The integral of (x^2-19)/(x^3-2x^2+x) is -19ln|x|+20ln|x-1|+(18)/(x-1)+C

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integral of (x^2-19)/(x^3-2x^2+x)

\int \frac{x ^{2} - 19}{x ^{3} - 2 x ^{2} + x} d x

- 19 ln ∣ x ∣ + 20 ln ∣ x - 1 ∣ + \frac{18}{x - 1} + C

Solution steps

\int \frac{x ^{2} - 19}{x ^{3} - 2 x ^{2} + x} d x

Take the partial fraction of

\frac{x ^{2} - 19}{x ^{3} - 2 x ^{2} + x} : - \frac{19}{x} + \frac{20}{x - 1} - \frac{18}{( x - 1 ) ^{2}}

= \int - \frac{19}{x} + \frac{20}{x - 1} - \frac{18}{( x - 1 ) ^{2}} 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{19}{x} d x + \int \frac{20}{x - 1} d x - \int \frac{18}{( x - 1 ) ^{2}} d x

\int \frac{19}{x} d x = 19 ln ∣ x ∣

\int \frac{20}{x - 1} d x = 20 ln ∣ x - 1 ∣

\int \frac{18}{( x - 1 ) ^{2}} d x = - \frac{18}{x - 1}

= - 19 ln ∣ x ∣ + 20 ln ∣ x - 1 ∣ - (- \frac{18}{x - 1})

Simplify

= - 19 ln ∣ x ∣ + 20 ln ∣ x - 1 ∣ + \frac{18}{x - 1}

Add a constant to the solution

= - 19 ln ∣ x ∣ + 20 ln ∣ x - 1 ∣ + \frac{18}{x - 1} + C

What is the integral of (x^2-19)/(x^3-2x^2+x) ?
The integral of (x^2-19)/(x^3-2x^2+x) is -19ln|x|+20ln|x-1|+(18)/(x-1)+C