What is the integral of (5x+1)/(sqrt(x^2-6x+5)) ?

The integral of (5x+1)/(sqrt(x^2-6x+5)) is 5sqrt(x^2-6x+5)+16ln(1/2 |sqrt(x^2-6x+5)+x-3|)+C

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integral of (5x+1)/(sqrt(x^2-6x+5))

\int \frac{5 x + 1}{x ^{2} - 6 x + 5} d x

5 x^{2} - 6 x + 5 + 16 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3) + C

Solution steps

\int \frac{5 x + 1}{x ^{2} - 6 x + 5} d x

Expand

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

= \int \frac{5 x}{x ^{2} - 6 x + 5} + \frac{1}{x ^{2} - 6 x + 5} 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{5 x}{x ^{2} - 6 x + 5} d x + \int \frac{1}{x ^{2} - 6 x + 5} d x

\int \frac{5 x}{x ^{2} - 6 x + 5} d x = 5 (x^{2} - 6 x + 5 + 3 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3))

\int \frac{1}{x ^{2} - 6 x + 5} d x = ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)

= 5 (x^{2} - 6 x + 5 + 3 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)) + ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)

Simplify

5 (x^{2} - 6 x + 5 + 3 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)) + ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3) : 5 x^{2} - 6 x + 5 + 16 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)

= 5 x^{2} - 6 x + 5 + 16 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3)

Add a constant to the solution

= 5 x^{2} - 6 x + 5 + 16 ln (\frac{1}{2} x^{2} - 6 x + 5 + x - 3) + C

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What is the integral of (5x+1)/(sqrt(x^2-6x+5)) ?
The integral of (5x+1)/(sqrt(x^2-6x+5)) is 5sqrt(x^2-6x+5)+16ln(1/2 |sqrt(x^2-6x+5)+x-3|)+C