What is the integral of (5x^2-12)/(x^2-6x+13) ?

The integral of (5x^2-12)/(x^2-6x+13) is 5x+13/2 arctan(1/2 (x-3))+15ln|x^2-6x+13|+C

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integral of (5x^2-12)/(x^2-6x+13)

\int \frac{5 x ^{2} - 12}{x ^{2} - 6 x + 13} d x

5 x + \frac{13}{2} arctan (\frac{1}{2} (x - 3)) + 15 ln x^{2} - 6 x + 13 + C

Solution steps

\int \frac{5 x ^{2} - 12}{x ^{2} - 6 x + 13} d x

Expand

\frac{5 x ^{2} - 12}{x ^{2} - 6 x + 13} : \frac{5 x ^{2}}{x ^{2} - 6 x + 13} - \frac{12}{x ^{2} - 6 x + 13}

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 ^{2}}{x ^{2} - 6 x + 13} d x - \int \frac{12}{x ^{2} - 6 x + 13} d x

\int \frac{5 x ^{2}}{x ^{2} - 6 x + 13} d x = 5 (3 ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + x)

\int \frac{12}{x ^{2} - 6 x + 13} d x = 6 arctan (\frac{1}{2} (x - 3))

= 5 (3 ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + x) - 6 arctan (\frac{1}{2} (x - 3))

Simplify

5 (3 ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + x) - 6 arctan (\frac{1}{2} (x - 3)) : 5 x + \frac{13}{2} arctan (\frac{1}{2} (x - 3)) + 15 ln x^{2} - 6 x + 13

= 5 x + \frac{13}{2} arctan (\frac{1}{2} (x - 3)) + 15 ln x^{2} - 6 x + 13

Add a constant to the solution

= 5 x + \frac{13}{2} arctan (\frac{1}{2} (x - 3)) + 15 ln x^{2} - 6 x + 13 + C

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What is the integral of (5x^2-12)/(x^2-6x+13) ?
The integral of (5x^2-12)/(x^2-6x+13) is 5x+13/2 arctan(1/2 (x-3))+15ln|x^2-6x+13|+C