integral of (x^5)/((x^2+4)^2)

解

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

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

解答ステップ

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

長除法

\frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} : x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16}

= \int x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} 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 x d x + \int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x

\int x d x = \frac{x ^{2}}{2}

\int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x = - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4}

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

定数を解答に追加する

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