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

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

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

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

2 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{5 x - 2}{x ^{2} - x} : \frac{2}{x} + \frac{3}{x - 1}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

= 2 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + C

y = x^{2}, y = 12 - x

\frac{d}{d x} (x + 13)

x \to \infty lim (ln ∣ x ∣)

y^{^{''}} + 81 y = 0, y (0) = 4, y^{^{'}} (0) = 7

y = (e^{2 x} - 2)^{2}, (0, 1)

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