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

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

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

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

2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)) + \frac{3}{7} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1) + 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}

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

\int \frac{3}{x ^{2} - 5 x + 1} d x = - \frac{3}{7} (ln \frac{2 x - 5}{21} + 1 - ln \frac{2 x - 5}{21} - 1)

= 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{2 x - 5}{21} + 1 - ln \frac{2 x - 5}{21} - 1)) - (- \frac{3}{7} (ln \frac{2 x - 5}{21} + 1 - ln \frac{2 x - 5}{21} - 1))

Simplify

2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{2 x - 5}{21} + 1 - ln \frac{2 x - 5}{21} - 1)) - (- \frac{3}{7} (ln \frac{2 x - 5}{21} + 1 - ln \frac{2 x - 5}{21} - 1)) : 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)) + \frac{3}{7} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)

= 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)) + \frac{3}{7} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)

Add a constant to the solution

= 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 4 - \frac{5}{4 21} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1)) + \frac{3}{7} (ln \frac{1}{21} (2 x - 5) + 1 - ln \frac{1}{21} (2 x - 5) - 1) + C

\int \frac{x}{1 + x x} d x

\int \frac{x ^{2} + 2}{( x + 1 ) ( x ^{2} + 4 )} d x

\int u^{\frac{- 1}{2}} d u

\int x ln (x) - x + 1 d x

\int - x + 2 - x^{2} d x

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