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

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

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

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

ln ∣ x ∣ - \frac{3}{x} - ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{2 x - 3}{x ^{3} - x ^{2}} : \frac{1}{x} + \frac{3}{x ^{2}} - \frac{1}{x - 1}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= ln ∣ x ∣ - \frac{3}{x} - ln ∣ x - 1 ∣

Add a constant to the solution

= ln ∣ x ∣ - \frac{3}{x} - ln ∣ x - 1 ∣ + C

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