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

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

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integral of ((3x^2-x+1))/(x^3-x^2)

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

\frac{1}{x} + 3 ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= - (- \frac{1}{x}) + 3 ln ∣ x - 1 ∣

Simplify

= \frac{1}{x} + 3 ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{1}{x} + 3 ln ∣ x - 1 ∣ + C

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