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

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

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

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

x^{2} - ln ∣ x ∣ + ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int 2 x d x = x^{2}

\int \frac{1}{x} d x = ln ∣ x ∣

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

= x^{2} - ln ∣ x ∣ + ln ∣ x - 1 ∣

Add a constant to the solution

= x^{2} - ln ∣ x ∣ + ln ∣ x - 1 ∣ + C

x \to \infty lim (\frac{3 ^{x - 3}}{2 ^{n + 3}})

x \to 2 - lim (x + 4)

3 y (x^{2} - 1) d x + (x^{3} + 8 y - 3 x) d y = 0

y = x^{2} - 5

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

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