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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x ^{2} + x + 1} + \frac{- x - 1}{( x ^{2} + x + 1 ) ^{2}} 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{1}{x ^{2} + x + 1} d x + \int \frac{- x - 1}{( x ^{2} + x + 1 ) ^{2}} d x

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

\int \frac{- x - 1}{( x ^{2} + x + 1 ) ^{2}} d x = - 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) - \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

= \frac{2}{3} arctan (\frac{2 x + 1}{3}) - 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) - \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

Simplify

\frac{2}{3} arctan (\frac{2 x + 1}{3}) - 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) - \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))) : \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) - \frac{2}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) + \frac{1}{2 ( x ^{2} + x + 1 )}

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

Add a constant to the solution

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

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