What is the integral of ((x+2))/((x^2+x+1)) ?

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

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integral of ((x+2))/((x^2+x+1))

integral

\frac{( x + 2 )}{( x ^{2} + x + 1 )}

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

Solution steps

\int \frac{x + 2}{x ^{2} + x + 1} d x

Expand

\frac{x + 2}{x ^{2} + x + 1} : \frac{x}{x ^{2} + x + 1} + \frac{2}{x ^{2} + x + 1}

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} + x + 1} d x + \int \frac{2}{x ^{2} + x + 1} d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

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