What is the integral of (10x-57)/(x^2-10x+41) ?

The integral of (10x-57)/(x^2-10x+41) is -7/4 arctan(1/4 (x-5))+5ln|x^2-10x+41|+C

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integral of (10x-57)/(x^2-10x+41)

\int \frac{10 x - 57}{x ^{2} - 10 x + 41} d x

- \frac{7}{4} arctan (\frac{1}{4} (x - 5)) + 5 ln x^{2} - 10 x + 41 + C

Solution steps

\int \frac{10 x - 57}{x ^{2} - 10 x + 41} d x

Apply Integration By Parts

: \frac{1}{4} arctan (\frac{1}{4} (x - 5)) (10 x - 57) - \int \frac{5}{2} arctan (\frac{1}{4} (x - 5)) d x

= \frac{1}{4} arctan (\frac{1}{4} (x - 5)) (10 x - 57) - \int \frac{5}{2} arctan (\frac{1}{4} (x - 5)) d x

\int \frac{5}{2} arctan (\frac{1}{4} (x - 5)) d x = \frac{5}{2} x arctan (\frac{1}{4} (x - 5)) - 5 ln x^{2} - 10 x + 41 - \frac{25}{2} arctan (\frac{1}{4} (x - 5))

= \frac{1}{4} arctan (\frac{1}{4} (x - 5)) (10 x - 57) - (\frac{5}{2} x arctan (\frac{1}{4} (x - 5)) - 5 ln x^{2} - 10 x + 41 - \frac{25}{2} arctan (\frac{1}{4} (x - 5)))

Simplify

\frac{1}{4} arctan (\frac{1}{4} (x - 5)) (10 x - 57) - (\frac{5}{2} x arctan (\frac{1}{4} (x - 5)) - 5 ln x^{2} - 10 x + 41 - \frac{25}{2} arctan (\frac{1}{4} (x - 5))) : - \frac{7}{4} arctan (\frac{1}{4} (x - 5)) + 5 ln x^{2} - 10 x + 41

= - \frac{7}{4} arctan (\frac{1}{4} (x - 5)) + 5 ln x^{2} - 10 x + 41

Add a constant to the solution

= - \frac{7}{4} arctan (\frac{1}{4} (x - 5)) + 5 ln x^{2} - 10 x + 41 + C

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What is the integral of (10x-57)/(x^2-10x+41) ?
The integral of (10x-57)/(x^2-10x+41) is -7/4 arctan(1/4 (x-5))+5ln|x^2-10x+41|+C