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

The integral of (x-2)/(sqrt(x^2-6x)) is sqrt(x^2-6x)+ln(1/3 |sqrt(x^2-6x)+x-3|)+C

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integral of (x-2)/(sqrt(x^2-6x))

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

x^{2} - 6 x + ln (\frac{1}{3} x^{2} - 6 x + x - 3) + C

Solution steps

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

Expand

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

= \int \frac{x}{x ^{2} - 6 x} - \frac{2}{x ^{2} - 6 x} 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{x}{x ^{2} - 6 x} d x - \int \frac{2}{x ^{2} - 6 x} d x

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x-2)/(sqrt(x^2-6x)) ?
The integral of (x-2)/(sqrt(x^2-6x)) is sqrt(x^2-6x)+ln(1/3 |sqrt(x^2-6x)+x-3|)+C