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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

乘开

\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

使用积分加法定则:

\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}}

u = x + 1

代回

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

化简

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

解答补常数

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

作图

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