What is the integral of (x-6)/(x^2-6x+8) ?

The integral of (x-6)/(x^2-6x+8) is 2ln|x-2|-ln|x-4|+C

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integral of (x-6)/(x^2-6x+8)

\int \frac{x - 6}{x ^{2} - 6 x + 8} d x

2 ln ∣ x - 2 ∣ - ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{x - 6}{x ^{2} - 6 x + 8} d x

Take the partial fraction of

\frac{x - 6}{x ^{2} - 6 x + 8} : \frac{2}{x - 2} - \frac{1}{x - 4}

= \int \frac{2}{x - 2} - \frac{1}{x - 4} 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 - 2} d x - \int \frac{1}{x - 4} d x

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

\int \frac{1}{x - 4} d x = ln ∣ x - 4 ∣

= 2 ln ∣ x - 2 ∣ - ln ∣ x - 4 ∣

Add a constant to the solution

= 2 ln ∣ x - 2 ∣ - ln ∣ x - 4 ∣ + C

\frac{d}{d x} (cos (1))

x \to 4 lim (\frac{4}{x ^{\frac{3}{2}}})

x \to - 3 lim (2 x^{2} + x^{3} + 5 x - 2)

\frac{6}{s} + 8 \cdot 2 \cdot \frac{1}{s ^{2} + 16}

\int yz e^{x z} d x

What is the integral of (x-6)/(x^2-6x+8) ?
The integral of (x-6)/(x^2-6x+8) is 2ln|x-2|-ln|x-4|+C