What is the integral of (x^2+2x+3)/(x^2+4x+13) ?

The integral of (x^2+2x+3)/(x^2+4x+13) is -ln| 1/9 (x+2)^2+1|-2arctan(1/3 (x+2))+x+C

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integral of (x^2+2x+3)/(x^2+4x+13)

\int \frac{x ^{2} + 2 x + 3}{x ^{2} + 4 x + 13} d x

- ln \frac{1}{9} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{3} (x + 2)) + x + C

Solution steps

\int \frac{x ^{2} + 2 x + 3}{x ^{2} + 4 x + 13} d x

\frac{x ^{2} + 2 x + 3}{x ^{2} + 4 x + 13} = \frac{- 2 x - 10}{x ^{2} + 4 x + 13} + 1

= \int \frac{- 2 x - 10}{x ^{2} + 4 x + 13} + 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{- 2 x - 10}{x ^{2} + 4 x + 13} d x + \int 1 d x

\int \frac{- 2 x - 10}{x ^{2} + 4 x + 13} d x = - ln \frac{1}{9} (x + 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x + 2)) - \frac{10}{3} arctan (\frac{1}{3} (x + 2))

\int 1 d x = x

= - ln \frac{1}{9} (x + 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x + 2)) - \frac{10}{3} arctan (\frac{1}{3} (x + 2)) + x

Add similar elements:

\frac{4}{3} arctan (\frac{1}{3} (x + 2)) - \frac{10}{3} arctan (\frac{1}{3} (x + 2)) = - 2 arctan (\frac{1}{3} (x + 2))

= - ln \frac{1}{9} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{3} (x + 2)) + x

Add a constant to the solution

= - ln \frac{1}{9} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{3} (x + 2)) + x + C

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What is the integral of (x^2+2x+3)/(x^2+4x+13) ?
The integral of (x^2+2x+3)/(x^2+4x+13) is -ln| 1/9 (x+2)^2+1|-2arctan(1/3 (x+2))+x+C