What is the integral of (5x+3)/(9x^2-12x+13) ?

The integral of (5x+3)/(9x^2-12x+13) is 5/18 ln|9x^2-12x+13|+19/27 arctan(x-2/3)+C

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integral of (5x+3)/(9x^2-12x+13)

\int \frac{5 x + 3}{9 x ^{2} - 12 x + 13} d x

\frac{5}{18} ln 9 x^{2} - 12 x + 13 + \frac{19}{27} arctan (x - \frac{2}{3}) + C

Solution steps

\int \frac{5 x + 3}{9 x ^{2} - 12 x + 13} d x

Expand

\frac{5 x + 3}{9 x ^{2} - 12 x + 13} : \frac{5 x}{9 x ^{2} - 12 x + 13} + \frac{3}{9 x ^{2} - 12 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}{9 x ^{2} - 12 x + 13} d x + \int \frac{3}{9 x ^{2} - 12 x + 13} d x

\int \frac{5 x}{9 x ^{2} - 12 x + 13} d x = \frac{5}{54} (3 ln 9 x^{2} - 12 x + 13 + 4 arctan (x - \frac{2}{3}))

\int \frac{3}{9 x ^{2} - 12 x + 13} d x = \frac{1}{3} arctan (x - \frac{2}{3})

= \frac{5}{54} (3 ln 9 x^{2} - 12 x + 13 + 4 arctan (x - \frac{2}{3})) + \frac{1}{3} arctan (x - \frac{2}{3})

Simplify

\frac{5}{54} (3 ln 9 x^{2} - 12 x + 13 + 4 arctan (x - \frac{2}{3})) + \frac{1}{3} arctan (x - \frac{2}{3}) : \frac{5}{18} ln 9 x^{2} - 12 x + 13 + \frac{19}{27} arctan (x - \frac{2}{3})

= \frac{5}{18} ln 9 x^{2} - 12 x + 13 + \frac{19}{27} arctan (x - \frac{2}{3})

Add a constant to the solution

= \frac{5}{18} ln 9 x^{2} - 12 x + 13 + \frac{19}{27} arctan (x - \frac{2}{3}) + C

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What is the integral of (5x+3)/(9x^2-12x+13) ?
The integral of (5x+3)/(9x^2-12x+13) is 5/18 ln|9x^2-12x+13|+19/27 arctan(x-2/3)+C