What is the integral of (x-3)/(-x^2+5x-6) ?

The integral of (x-3)/(-x^2+5x-6) is -ln|x-2|+C

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integral of (x-3)/(-x^2+5x-6)

\int \frac{x - 3}{- x ^{2} + 5 x - 6} d x

- ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x - 3}{- x ^{2} + 5 x - 6} d x

Simplify

\frac{x - 3}{- x ^{2} + 5 x - 6} : - \frac{1}{x - 2}

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = x - 2

= - ln ∣ x - 2 ∣

Add a constant to the solution

= - ln ∣ x - 2 ∣ + C

What is the integral of (x-3)/(-x^2+5x-6) ?
The integral of (x-3)/(-x^2+5x-6) is -ln|x-2|+C