What is the integral of (5x-3)/(sqrt(x^2-2x+1)) ?

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

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integral of (5x-3)/(sqrt(x^2-2x+1))

\int \frac{5 x - 3}{x ^{2} - 2 x + 1} d x

5 x - 5 + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{5 x - 3}{x ^{2} - 2 x + 1} d x

Expand

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

= \int \frac{5 x}{x ^{2} - 2 x + 1} - \frac{3}{x ^{2} - 2 x + 1} 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} - 2 x + 1} d x - \int \frac{3}{x ^{2} - 2 x + 1} d x

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

\int \frac{3}{x ^{2} - 2 x + 1} d x = 3 ln ∣ x - 1 ∣

= 5 (x - 1 + ln ∣ x - 1 ∣) - 3 ln ∣ x - 1 ∣

Simplify

5 (x - 1 + ln ∣ x - 1 ∣) - 3 ln ∣ x - 1 ∣ : 5 x - 5 + 2 ln ∣ x - 1 ∣

= 5 x - 5 + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= 5 x - 5 + 2 ln ∣ x - 1 ∣ + C

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