What is the integral of (2x^3-5x^2+4x-4)/(x^2-x) ?

The integral of (2x^3-5x^2+4x-4)/(x^2-x) is x^2-3x+2ln(2|x-1|)+4ln|x|-5ln|x-1|+2+C

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integral of (2x^3-5x^2+4x-4)/(x^2-x)

\int \frac{2 x ^{3} - 5 x ^{2} + 4 x - 4}{x ^{2} - x} d x

x^{2} - 3 x + 2 ln (2 ∣ x - 1 ∣) + 4 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ + 2 + C

Solution steps

\int \frac{2 x ^{3} - 5 x ^{2} + 4 x - 4}{x ^{2} - x} d x

Expand

\frac{2 x ^{3} - 5 x ^{2} + 4 x - 4}{x ^{2} - x} : \frac{2 x ^{3}}{x ^{2} - x} - \frac{5 x ^{2}}{x ^{2} - x} + \frac{4 x}{x ^{2} - x} - \frac{4}{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{2 x ^{3}}{x ^{2} - x} d x - \int \frac{5 x ^{2}}{x ^{2} - x} d x + \int \frac{4 x}{x ^{2} - x} d x - \int \frac{4}{x ^{2} - x} d x

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

\int \frac{5 x ^{2}}{x ^{2} - x} d x = 5 (x - 1 + ln ∣ x - 1 ∣)

\int \frac{4 x}{x ^{2} - x} d x = 4 ln ∣ x - 1 ∣

\int \frac{4}{x ^{2} - x} d x = 4 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

= 2 ln (2 ∣ x - 1 ∣) + (x - 1) (x + 3) - 5 (x - 1 + ln ∣ x - 1 ∣) + 4 ln ∣ x - 1 ∣ - 4 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

Simplify

2 ln (2 ∣ x - 1 ∣) + (x - 1) (x + 3) - 5 (x - 1 + ln ∣ x - 1 ∣) + 4 ln ∣ x - 1 ∣ - 4 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) : x^{2} - 3 x + 2 ln (2 ∣ x - 1 ∣) + 4 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ + 2

= x^{2} - 3 x + 2 ln (2 ∣ x - 1 ∣) + 4 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ + 2

Add a constant to the solution

= x^{2} - 3 x + 2 ln (2 ∣ x - 1 ∣) + 4 ln ∣ x ∣ - 5 ln ∣ x - 1 ∣ + 2 + C

\int (e^{x} + x^{2}) d x

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

\int 5 x^{3} x^{2} + 2 d x

What is the integral of (2x^3-5x^2+4x-4)/(x^2-x) ?
The integral of (2x^3-5x^2+4x-4)/(x^2-x) is x^2-3x+2ln(2|x-1|)+4ln|x|-5ln|x-1|+2+C