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人気のある微分積分 >

integral of x/((x+1)^8)

解

\int \frac{x}{( x + 1 ) ^{8}} d x

解

- \frac{1}{6 ( x + 1 ) ^{6}} + \frac{1}{7 ( x + 1 ) ^{7}} + C

解答ステップ

\int \frac{x}{( x + 1 ) ^{8}} d x

u置換積分法を適用する

= \int \frac{u - 1}{u ^{8}} d u

拡張

\frac{u - 1}{u ^{8}} : \frac{1}{u ^{7}} - \frac{1}{u ^{8}}

= \int \frac{1}{u ^{7}} - \frac{1}{u ^{8}} 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

= \int \frac{1}{u ^{7}} d u - \int \frac{1}{u ^{8}} d u

\int \frac{1}{u ^{7}} d u = - \frac{1}{6 u ^{6}}

\int \frac{1}{u ^{8}} d u = - \frac{1}{7 u ^{7}}

= - \frac{1}{6 u ^{6}} - (- \frac{1}{7 u ^{7}})

代用を戻す

u = x + 1

= - \frac{1}{6 ( x + 1 ) ^{6}} - (- \frac{1}{7 ( x + 1 ) ^{7}})

簡素化

= - \frac{1}{6 ( x + 1 ) ^{6}} + \frac{1}{7 ( x + 1 ) ^{7}}

定数を解答に追加する

= - \frac{1}{6 ( x + 1 ) ^{6}} + \frac{1}{7 ( x + 1 ) ^{7}} + C

グラフ

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