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人気のある微分積分 >

integral of (x^2)/((1-x)^{100)}

解

\int \frac{x ^{2}}{( 1 - x ) ^{100}} d x

解

\frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}} + C

解答ステップ

\int \frac{x ^{2}}{( 1 - x ) ^{100}} d x

u置換積分法を適用する

= \int - \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

定数を除く:

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

= - \int \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

拡張

\frac{( - u + 1 ) ^{2}}{u ^{100}} : \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}}

= - \int \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}} 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 ^{98}} d u - \int \frac{2}{u ^{99}} d u + \int \frac{1}{u ^{100}} d u)

\int \frac{1}{u ^{98}} d u = - \frac{1}{97 u ^{97}}

\int \frac{2}{u ^{99}} d u = - \frac{1}{49 u ^{98}}

\int \frac{1}{u ^{100}} d u = - \frac{1}{99 u ^{99}}

= - (- \frac{1}{97 u ^{97}} - (- \frac{1}{49 u ^{98}}) - \frac{1}{99 u ^{99}})

代用を戻す

u = 1 - x

= - (- \frac{1}{97 ( 1 - x ) ^{97}} - (- \frac{1}{49 ( 1 - x ) ^{98}}) - \frac{1}{99 ( 1 - x ) ^{99}})

簡素化

= \frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}}

定数を解答に追加する

= \frac{1}{97 ( 1 - x ) ^{97}} - \frac{1}{49 ( 1 - x ) ^{98}} + \frac{1}{99 ( 1 - x ) ^{99}} + C

グラフ

人気の例

integral of xln((1+x)/(1-x))

\int x ln (\frac{1 + x}{1 - x}) d x

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\int e^{2 x} arctan (e^{x} - 1) d x

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\int sin^{2} (x) tan (x) d x

integral of 1/(x^2-8x+25)

\int \frac{1}{x ^{2} - 8 x + 25} d x

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

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