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

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

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

\int (\frac{3 x ^{2} - 2 x}{x ^{3} - x ^{2}}) d x

ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

Solution steps

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

Simplify

\frac{3 x ^{2} - 2 x}{x ^{3} - x ^{2}} : \frac{3 x - 2}{x ( x - 1 )}

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

Expand

\frac{3 x - 2}{x ( x - 1 )} : \frac{3 x}{x ( x - 1 )} - \frac{2}{x ( x - 1 )}

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{3 x}{x ( x - 1 )} d x - \int \frac{2}{x ( x - 1 )} d x

\int \frac{3 x}{x ( x - 1 )} d x = 3 ln ∣ x - 1 ∣

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

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

Simplify

3 ln ∣ x - 1 ∣ - 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) : ln ∣ x - 1 ∣ + 2 ln ∣ x ∣

= ln ∣ x - 1 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ + 2 ln ∣ x ∣ + C

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