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

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

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

\int \frac{( 3 )}{( x ^{2} - x )} d x

3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

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

= 3 \cdot \int - \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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

Add a constant to the solution

= 3 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + C

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