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人気のある微分積分 >

integral of (x+1)^{-8}x

解

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

解

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

解答ステップ

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

簡素化

(x + 1)^{- 8} x : \frac{x}{( x + 1 ) ^{8}}

= \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

人気の例

limit as x approaches 8 of sqrt(7x-4)

x \to 8 lim (7 x - 4)

sum from n=0 to infinity of 1.002^n

n = 0 \sum \infty 1.00 2^{n}

integral from 0 to 2 of xe^{x^2}

\int_{0}^{2} x e^{x^{2}} d x

integral of (3x^2-7)/(x^3)

\int \frac{3 x ^{2} - 7}{x ^{3}} d x

面積 y=x+2,y=-1,x=2,x=5

a re a y = x + 2, y = - 1, x = 2, x = 5