What is the integral of (4x-6)/(x^2-2x) ?

The integral of (4x-6)/(x^2-2x) is 3ln|x|+ln|x-2|+C

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integral of (4x-6)/(x^2-2x)

\int \frac{4 x - 6}{x ^{2} - 2 x} d x

3 ln ∣ x ∣ + ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{4 x - 6}{x ^{2} - 2 x} d x

Take the partial fraction of

\frac{4 x - 6}{x ^{2} - 2 x} : \frac{3}{x} + \frac{1}{x - 2}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= 3 ln ∣ x ∣ + ln ∣ x - 2 ∣

Add a constant to the solution

= 3 ln ∣ x ∣ + ln ∣ x - 2 ∣ + C

\frac{d ^{2}}{d x ^{2}} (e^{x^{2}})

\int x \cdot e^{- 4 x} d x

\frac{d}{d y} (x^{8 y})

\int 2 π x sin (x^{2}) d x

\frac{d y}{d x}, x^{\frac{1}{2}} + y^{\frac{1}{2}} = 9

What is the integral of (4x-6)/(x^2-2x) ?
The integral of (4x-6)/(x^2-2x) is 3ln|x|+ln|x-2|+C