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

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

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

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

x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{1}{21} (2 x - 5) + C

Solution steps

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

Expand

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

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

\int \frac{x}{x ^{2} - 5 x + 1} d x = x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{4 x ^{2} - 20 x + 4}{21} + \frac{2}{21} x - \frac{5}{21}

\int \frac{3}{x ^{2} - 5 x + 1} d x = 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21}

= x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{4 x ^{2} - 20 x + 4}{21} + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21}

Simplify

x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{4 x ^{2} - 20 x + 4}{21} + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21} : x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{1}{21} (2 x - 5)

= x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{1}{21} (2 x - 5)

Add a constant to the solution

= x^{2} - 5 x + 1 + \frac{5}{2} ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{2}{21} x - \frac{5}{21} - 3 ln \frac{1}{21} (4 x^{2} - 20 x + 4) + \frac{1}{21} (2 x - 5) + C

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