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

解

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

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

解答ステップ

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

拡張

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\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 )}

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

簡素化

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

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

定数を解答に追加する

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