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

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

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

integral

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

3 (- ln ∣ x ∣ + ln ∣ 2 x - 1 ∣) + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= 3 (- ln ∣ x ∣ + ln ∣ 2 x - 1 ∣)

Add a constant to the solution

= 3 (- ln ∣ x ∣ + ln ∣ 2 x - 1 ∣) + C

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