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人気のある微分積分 >

integral of (2x)/((2-x)^6)

解

\int \frac{2 x}{( 2 - x ) ^{6}} d x

解

- 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}}) + C

解答ステップ

\int \frac{2 x}{( 2 - x ) ^{6}} 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}{( 2 - x ) ^{6}} d x

u置換積分法を適用する

= 2 \cdot \int - \frac{- u + 2}{u ^{6}} d u

定数を除く:

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

= 2 (- \int \frac{- u + 2}{u ^{6}} d u)

拡張

\frac{- u + 2}{u ^{6}} : - \frac{1}{u ^{5}} + \frac{2}{u ^{6}}

= 2 (- \int - \frac{1}{u ^{5}} + \frac{2}{u ^{6}} d u)

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}{u ^{5}} d u + \int \frac{2}{u ^{6}} d u))

\int \frac{1}{u ^{5}} d u = - \frac{1}{4 u ^{4}}

\int \frac{2}{u ^{6}} d u = - \frac{2}{5 u ^{5}}

= 2 (- (- (- \frac{1}{4 u ^{4}}) - \frac{2}{5 u ^{5}}))

代用を戻す

u = 2 - x

= 2 (- (- (- \frac{1}{4 ( 2 - x ) ^{4}}) - \frac{2}{5 ( 2 - x ) ^{5}}))

簡素化

= - 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}})

定数を解答に追加する

= - 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}}) + C

グラフ

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