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

The integral of (x-3)/(x^2+3x+3) is 1/2 ln|4x^2+12x+12|-3sqrt(3)arctan(2/(sqrt(3))x+sqrt(3))+C

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

\int \frac{x - 3}{x ^{2} + 3 x + 3} d x

\frac{1}{2} ln 4 x^{2} + 12 x + 12 - 33 arctan (\frac{2}{3} x + 3) + C

Solution steps

\int \frac{x - 3}{x ^{2} + 3 x + 3} d x

Expand

\frac{x - 3}{x ^{2} + 3 x + 3} : \frac{x}{x ^{2} + 3 x + 3} - \frac{3}{x ^{2} + 3 x + 3}

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} + 3 x + 3} d x - \int \frac{3}{x ^{2} + 3 x + 3} d x

\int \frac{x}{x ^{2} + 3 x + 3} d x = \frac{1}{2} ln 4 x^{2} + 12 x + 12 - 3 arctan (\frac{2}{3} x + 3)

\int \frac{3}{x ^{2} + 3 x + 3} d x = 23 arctan (\frac{2}{3} x + 3)

= \frac{1}{2} ln 4 x^{2} + 12 x + 12 - 3 arctan (\frac{2}{3} x + 3) - 23 arctan (\frac{2}{3} x + 3)

Simplify

= \frac{1}{2} ln 4 x^{2} + 12 x + 12 - 33 arctan (\frac{2}{3} x + 3)

Add a constant to the solution

= \frac{1}{2} ln 4 x^{2} + 12 x + 12 - 33 arctan (\frac{2}{3} x + 3) + C

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What is the integral of (x-3)/(x^2+3x+3) ?
The integral of (x-3)/(x^2+3x+3) is 1/2 ln|4x^2+12x+12|-3sqrt(3)arctan(2/(sqrt(3))x+sqrt(3))+C