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

y^{^{''}} + 49 y = cos (10 t)

ln (x^{3} + 5 x + 3)

x \to 1 lim (x sin (\frac{1}{x}))

y = x^{4} - 3 x^{2} + 2, (1, 0)

x \to 2 - lim (\frac{∣ x - x ∣}{x - 2})

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