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

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

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

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

\frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + 2 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

\int \frac{x ^{5}}{x ^{2} - x} d x = \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + ln ∣ x - 1 ∣

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

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

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

Simplify

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

= \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + 2 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + 2 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ + C

y = \frac{1}{5} x^{2} - \frac{6}{5} x + \frac{7}{5}, (- 5, 5)

x \to - 1 lim (e^{x - 2})

\int x sin (4 - x) d x

\int ln^{2} (x) d x

y^{^{''}} - 4 y^{^{'}} + 4 y = 12 t e^{2 t} + 6 e^{2 t} + 24 - 12 t

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