integral of (2x+1)/(x^6+18x^4+81x^2)

解

\int \frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} d x

\frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )} + C

解答ステップ

\int \frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} d x

以下の部分分数を得る:

\frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} : \frac{2}{81 x} + \frac{1}{81 x ^{2}} + \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} + \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}}

= \int \frac{2}{81 x} + \frac{1}{81 x ^{2}} + \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} + \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x

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{2}{81 x} d x + \int \frac{1}{81 x ^{2}} d x + \int \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} d x + \int \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x

\int \frac{2}{81 x} d x = \frac{2}{81} ln ∣ x ∣

\int \frac{1}{81 x ^{2}} d x = - \frac{1}{81 x}

\int \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} d x = \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}))

\int \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x = \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))))

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81 x} + \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})) + \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))))

簡素化

\frac{2}{81} ln ∣ x ∣ - \frac{1}{81 x} + \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})) + \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3})))) : \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )}

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )}

定数を解答に追加する

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )} + C