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

The integral of (-x+8)/(x^2-6x+8) is -1/2 ln|x^2-6x+8|-5/2 ln|x-2|+5/2 ln|x-4|+C

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

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

- \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{x}{x ^{2} - 6 x + 8} d x = \frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)

\int \frac{8}{x ^{2} - 6 x + 8} d x = 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣)

= - (\frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)) + 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣)

Simplify

- (\frac{1}{2} ln x^{2} - 6 x + 8 - 3 (\frac{1}{2} ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x - 4 ∣)) + 8 (- \frac{1}{2} ln ∣ x - 2 ∣ + \frac{1}{2} ln ∣ x - 4 ∣) : - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣

= - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣

Add a constant to the solution

= - \frac{1}{2} ln x^{2} - 6 x + 8 - \frac{5}{2} ln ∣ x - 2 ∣ + \frac{5}{2} ln ∣ x - 4 ∣ + C

What is the integral of (-x+8)/(x^2-6x+8) ?
The integral of (-x+8)/(x^2-6x+8) is -1/2 ln|x^2-6x+8|-5/2 ln|x-2|+5/2 ln|x-4|+C