What is the integral of (3x-4)/(x^2-2x) ?

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

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integral of (3x-4)/(x^2-2x)

\int \frac{3 x - 4}{x ^{2} - 2 x} d x

2 ln ∣ x ∣ + ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{3 x - 4}{x ^{2} - 2 x} d x

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ + ln ∣ x - 2 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ + ln ∣ x - 2 ∣ + C

\int 3 x^{2} d x

\int 4 x^{3} - 3 x^{2} + 5 d x

\int (3 x^{2} y^{- 2} + 2 x y^{- 1}) d x

\frac{d}{d x} (\frac{( x + 1 ) ( 3 x - 7 )}{3 x + 1})

\int_{- 2}^{2} (x^{2} - 2)^{2} d x

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