What is the integral of (8x^2-13x+2)/(x^3-3x^2+2x) ?

The integral of (8x^2-13x+2)/(x^3-3x^2+2x) is 3ln|x-1|+4ln|x-2|+ln|x|+C

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integral of (8x^2-13x+2)/(x^3-3x^2+2x)

\int \frac{8 x ^{2} - 13 x + 2}{x ^{3} - 3 x ^{2} + 2 x} d x

3 ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

Solution steps

\int \frac{8 x ^{2} - 13 x + 2}{x ^{3} - 3 x ^{2} + 2 x} d x

Expand

\frac{8 x ^{2} - 13 x + 2}{x ^{3} - 3 x ^{2} + 2 x} : \frac{8 x ^{2}}{x ^{3} - 3 x ^{2} + 2 x} - \frac{13 x}{x ^{3} - 3 x ^{2} + 2 x} + \frac{2}{x ^{3} - 3 x ^{2} + 2 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{8 x ^{2}}{x ^{3} - 3 x ^{2} + 2 x} d x - \int \frac{13 x}{x ^{3} - 3 x ^{2} + 2 x} d x + \int \frac{2}{x ^{3} - 3 x ^{2} + 2 x} d x

\int \frac{8 x ^{2}}{x ^{3} - 3 x ^{2} + 2 x} d x = 8 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣)

\int \frac{13 x}{x ^{3} - 3 x ^{2} + 2 x} d x = 13 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣)

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

= 8 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - 13 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) + ln ∣ x ∣ - 2 ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣

Simplify

8 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - 13 (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) + ln ∣ x ∣ - 2 ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣ : 3 ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + ln ∣ x ∣

= 3 ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + ln ∣ x ∣

Add a constant to the solution

= 3 ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

\int \frac{sin ^{7} ( 8 x )}{cos ^{4} ( 8 x )} d x

\frac{\partial}{\partial x} (ln (x^{6} + y^{3}))

sin (x) \frac{π}{4}

\int sec^{2} (11 x) d x

e^{- 9 t}

What is the integral of (8x^2-13x+2)/(x^3-3x^2+2x) ?
The integral of (8x^2-13x+2)/(x^3-3x^2+2x) is 3ln|x-1|+4ln|x-2|+ln|x|+C