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人気のある微分積分 >

integral of (x^6-x^3+1)/(x^4+9x^2)

解

\int \frac{x ^{6} - x ^{3} + 1}{x ^{4} + 9 x ^{2}} d x

解

\frac{x ^{3}}{3} - 9 x - \frac{1}{2} ln x^{2} + 9 + \frac{728}{27} arctan (\frac{x}{3}) - \frac{1}{9 x} + C

解答ステップ

\int \frac{x ^{6} - x ^{3} + 1}{x ^{4} + 9 x ^{2}} d x

因数

\frac{x ^{6} - x ^{3} + 1}{x ^{4} + 9 x ^{2}}

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

拡張

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

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

\int \frac{x ^{6}}{x ^{2} ( x ^{2} + 9 )} d x = 27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3}))

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

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

= 27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3})) - \frac{1}{2} ln x^{2} + 9 - \frac{1}{9 x} - \frac{1}{27} arctan (\frac{x}{3})

簡素化

27 (\frac{x ^{3}}{81} - \frac{x}{3} + arctan (\frac{x}{3})) - \frac{1}{2} ln x^{2} + 9 - \frac{1}{9 x} - \frac{1}{27} arctan (\frac{x}{3}) : \frac{x ^{3}}{3} - 9 x - \frac{1}{2} ln x^{2} + 9 + \frac{728}{27} arctan (\frac{x}{3}) - \frac{1}{9 x}

= \frac{x ^{3}}{3} - 9 x - \frac{1}{2} ln x^{2} + 9 + \frac{728}{27} arctan (\frac{x}{3}) - \frac{1}{9 x}

定数を解答に追加する

= \frac{x ^{3}}{3} - 9 x - \frac{1}{2} ln x^{2} + 9 + \frac{728}{27} arctan (\frac{x}{3}) - \frac{1}{9 x} + C

グラフ

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