What is the integral of (4x-10)/(x^2-6x+8) ?

The integral of (4x-10)/(x^2-6x+8) is ln|x-2|+3ln|x-4|+C

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integral of (4x-10)/(x^2-6x+8)

\int \frac{4 x - 10}{x ^{2} - 6 x + 8} d x

ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{4 x - 10}{x ^{2} - 6 x + 8} d x

Take the partial fraction of

\frac{4 x - 10}{x ^{2} - 6 x + 8} : \frac{1}{x - 2} + \frac{3}{x - 4}

= \int \frac{1}{x - 2} + \frac{3}{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{1}{x - 2} d x + \int \frac{3}{x - 4} d x

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

\int \frac{3}{x - 4} d x = 3 ln ∣ x - 4 ∣

= ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣

Add a constant to the solution

= ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣ + C

\int \frac{x + 2}{x + 5} d x

f (t) = t^{6}

f (x) = 5 x^{4} - 5 x^{2} + 7

\int \frac{1}{x ^{2} + 8 x + 16} d x

\frac{d}{d x} (lo g_{5} (e^{x}))

What is the integral of (4x-10)/(x^2-6x+8) ?
The integral of (4x-10)/(x^2-6x+8) is ln|x-2|+3ln|x-4|+C