What is the integral of (12x-2)/(6x^2-2x) ?

The integral of (12x-2)/(6x^2-2x) is ln|3x-1|+ln|x|+C

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integral of (12x-2)/(6x^2-2x)

\int \frac{12 x - 2}{6 x ^{2} - 2 x} d x

ln ∣ 3 x - 1 ∣ + ln ∣ x ∣ + C

Solution steps

\int \frac{12 x - 2}{6 x ^{2} - 2 x} d x

Expand

\frac{12 x - 2}{6 x ^{2} - 2 x} : \frac{12 x}{6 x ^{2} - 2 x} - \frac{2}{6 x ^{2} - 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{12 x}{6 x ^{2} - 2 x} d x - \int \frac{2}{6 x ^{2} - 2 x} d x

\int \frac{12 x}{6 x ^{2} - 2 x} d x = 2 ln ∣ 3 x - 1 ∣

\int \frac{2}{6 x ^{2} - 2 x} d x = - ln ∣ x ∣ + ln ∣ 3 x - 1 ∣

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

Simplify

= ln ∣ 3 x - 1 ∣ + ln ∣ x ∣

Add a constant to the solution

= ln ∣ 3 x - 1 ∣ + ln ∣ x ∣ + C

\frac{d}{d x} ((x^{3} + 3) (x + 2))

\frac{d}{d x} (50 x - x^{2})

\frac{d y}{d x} = k (y - 70)

f (x) = 3 x 2 - 5 x, at x = - 1

\frac{\partial}{\partial x} (- 4 x y)

What is the integral of (12x-2)/(6x^2-2x) ?
The integral of (12x-2)/(6x^2-2x) is ln|3x-1|+ln|x|+C