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人気のある微分積分 >

integral of-(2x)/((x^2-2)^3)

解

\int - \frac{2 x}{( x ^{2} - 2 ) ^{3}} d x

解

- 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}}) + C

解答ステップ

\int - \frac{2 x}{( x ^{2} - 2 ) ^{3}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{x}{( x ^{2} - 2 ) ^{3}} d x

以下の部分分数を得る:

\frac{x}{( x ^{2} - 2 ) ^{3}} : \frac{1}{32 2 ( x + 2 ) ^{2}} + \frac{1}{16 ( x + 2 ) ^{3}} - \frac{1}{32 2 ( x - 2 ) ^{2}} + \frac{1}{16 ( x - 2 ) ^{3}}

= - 2 \cdot \int \frac{1}{32 2 ( x + 2 ) ^{2}} + \frac{1}{16 ( x + 2 ) ^{3}} - \frac{1}{32 2 ( x - 2 ) ^{2}} + \frac{1}{16 ( x - 2 ) ^{3}} 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

= - 2 (\int \frac{1}{32 2 ( x + 2 ) ^{2}} d x + \int \frac{1}{16 ( x + 2 ) ^{3}} d x - \int \frac{1}{32 2 ( x - 2 ) ^{2}} d x + \int \frac{1}{16 ( x - 2 ) ^{3}} d x)

\int \frac{1}{32 2 ( x + 2 ) ^{2}} d x = - \frac{1}{32 2 ( x + 2 )}

\int \frac{1}{16 ( x + 2 ) ^{3}} d x = - \frac{1}{32 ( x + 2 ) ^{2}}

\int \frac{1}{32 2 ( x - 2 ) ^{2}} d x = - \frac{1}{32 2 ( x - 2 )}

\int \frac{1}{16 ( x - 2 ) ^{3}} d x = - \frac{1}{32 ( x - 2 ) ^{2}}

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} - (- \frac{1}{32 2 ( x - 2 )}) - \frac{1}{32 ( x - 2 ) ^{2}})

簡素化

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}})

定数を解答に追加する

= - 2 (- \frac{1}{32 2 ( x + 2 )} - \frac{1}{32 ( x + 2 ) ^{2}} + \frac{1}{32 2 ( x - 2 )} - \frac{1}{32 ( x - 2 ) ^{2}}) + C

グラフ

人気の例

integral of (2x^2-x+3)/x

\int \frac{2 x ^{2} - x + 3}{x} d x

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h \to 0 lim (\frac{sin ( h )}{h})

y^{^{''}} - y = t

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\int \frac{- 1}{sin ( x )} d x

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