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人気のある微分積分 >

integral of (x^2)/((x+1)^{50)}

解

\int \frac{x ^{2}}{( x + 1 ) ^{50}} d x

解

- \frac{1}{47 ( x + 1 ) ^{47}} + \frac{1}{24 ( x + 1 ) ^{48}} - \frac{1}{49 ( x + 1 ) ^{49}} + C

解答ステップ

\int \frac{x ^{2}}{( x + 1 ) ^{50}} d x

u置換積分法を適用する

= \int \frac{( u - 1 ) ^{2}}{u ^{50}} d u

拡張

\frac{( u - 1 ) ^{2}}{u ^{50}} : \frac{1}{u ^{48}} - \frac{2}{u ^{49}} + \frac{1}{u ^{50}}

= \int \frac{1}{u ^{48}} - \frac{2}{u ^{49}} + \frac{1}{u ^{50}} 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 ^{48}} d u - \int \frac{2}{u ^{49}} d u + \int \frac{1}{u ^{50}} d u

\int \frac{1}{u ^{48}} d u = - \frac{1}{47 u ^{47}}

\int \frac{2}{u ^{49}} d u = - \frac{1}{24 u ^{48}}

\int \frac{1}{u ^{50}} d u = - \frac{1}{49 u ^{49}}

= - \frac{1}{47 u ^{47}} - (- \frac{1}{24 u ^{48}}) - \frac{1}{49 u ^{49}}

代用を戻す

u = x + 1

= - \frac{1}{47 ( x + 1 ) ^{47}} - (- \frac{1}{24 ( x + 1 ) ^{48}}) - \frac{1}{49 ( x + 1 ) ^{49}}

簡素化

= - \frac{1}{47 ( x + 1 ) ^{47}} + \frac{1}{24 ( x + 1 ) ^{48}} - \frac{1}{49 ( x + 1 ) ^{49}}

定数を解答に追加する

= - \frac{1}{47 ( x + 1 ) ^{47}} + \frac{1}{24 ( x + 1 ) ^{48}} - \frac{1}{49 ( x + 1 ) ^{49}} + C

グラフ

人気の例

integral of (2x-1)ln(17x)

\int (2 x - 1) ln (17 x) d x

面積 2/x ,4x, 4/x

a re a \frac{2}{x}, 4 x, \frac{4}{x}

テイラー 5x

t a y l or 5 x

x y^{^{'}} - 4 y = x^{6} e^{x}

limit as x approaches infinity of (x^{99})/(e^x)

x \to \infty lim (\frac{x ^{99}}{e ^{x}})