integral of (x^4-2x^3-27)/(x(x^2+9)^2)

解

\int \frac{x ^{4} - 2 x ^{3} - 27}{x ( x ^{2} + 9 ) ^{2}} d x

\frac{2}{3} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) - \frac{1}{3} ln (x) + \frac{3}{x ^{2} + 9} + \frac{x}{x ^{2} + 9} + C

解答ステップ

\int \frac{x ^{4} - 2 x ^{3} - 27}{x ( x ^{2} + 9 ) ^{2}} d x

拡張

\frac{x ^{4} - 2 x ^{3} - 27}{x ( x ^{2} + 9 ) ^{2}} : \frac{x ^{4}}{x ( x ^{2} + 9 ) ^{2}} - \frac{2 x ^{3}}{x ( x ^{2} + 9 ) ^{2}} - \frac{27}{x ( x ^{2} + 9 ) ^{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 ^{4}}{x ( x ^{2} + 9 ) ^{2}} d x - \int \frac{2 x ^{3}}{x ( x ^{2} + 9 ) ^{2}} d x - \int \frac{27}{x ( x ^{2} + 9 ) ^{2}} d x

\int \frac{x ^{4}}{x ( x ^{2} + 9 ) ^{2}} d x = \frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9})

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

\int \frac{27}{x ( x ^{2} + 9 ) ^{2}} d x = \frac{1}{3} ln (x) - \frac{1}{6} ln x^{2} + 9 + \frac{3}{2 ( x ^{2} + 9 )}

= \frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9}) - (\frac{1}{3} arctan (\frac{x}{3}) - \frac{x}{9 + x ^{2}}) - (\frac{1}{3} ln (x) - \frac{1}{6} ln x^{2} + 9 + \frac{3}{2 ( x ^{2} + 9 )})

簡素化

\frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9}) - (\frac{1}{3} arctan (\frac{x}{3}) - \frac{x}{9 + x ^{2}}) - (\frac{1}{3} ln (x) - \frac{1}{6} ln x^{2} + 9 + \frac{3}{2 ( x ^{2} + 9 )}) : \frac{2}{3} ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}) - \frac{1}{3} ln (x) + \frac{3}{x ^{2} + 9} + \frac{x}{x ^{2} + 9}

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

定数を解答に追加する

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