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

The integral of (3x^3-3x^2+1)/(x^2-x) is 3/2 x^2+3ln(2|x-1|)-2ln|x-1|-ln|x|-9/2+3+C

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

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

\frac{3}{2} x^{2} + 3 ln (2 ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣ - ln ∣ x ∣ - \frac{9}{2} + 3 + C

Solution steps

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

Expand

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

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

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

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

= \frac{3}{2} (2 ln (2 ∣ x - 1 ∣) + (x - 1) (x + 3)) - 3 (x - 1 + ln ∣ x - 1 ∣) - ln ∣ x ∣ + ln ∣ x - 1 ∣

Simplify

\frac{3}{2} (2 ln (2 ∣ x - 1 ∣) + (x - 1) (x + 3)) - 3 (x - 1 + ln ∣ x - 1 ∣) - ln ∣ x ∣ + ln ∣ x - 1 ∣ : \frac{3}{2} x^{2} + 3 ln (2 ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣ - ln ∣ x ∣ - \frac{9}{2} + 3

= \frac{3}{2} x^{2} + 3 ln (2 ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣ - ln ∣ x ∣ - \frac{9}{2} + 3

Add a constant to the solution

= \frac{3}{2} x^{2} + 3 ln (2 ∣ x - 1 ∣) - 2 ln ∣ x - 1 ∣ - ln ∣ x ∣ - \frac{9}{2} + 3 + C

\int x arcsin (d) x d x

\frac{d}{d x} (- 7 ln (3 x + 4))

\int x (x^{2} + 3)^{4} d x

x \to 0 lim (8 x)

\frac{\partial}{\partial x} (3 x^{2} - 5 y)

What is the integral of (3x^3-3x^2+1)/(x^2-x) ?
The integral of (3x^3-3x^2+1)/(x^2-x) is 3/2 x^2+3ln(2|x-1|)-2ln|x-1|-ln|x|-9/2+3+C