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受欢迎的微积分 >

integral of (2x)/((2-x)^6)

解答

\int \frac{2 x}{( 2 - x ) ^{6}} d x

解答

- 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}}) + C

求解步骤

\int \frac{2 x}{( 2 - x ) ^{6}} d x

提出常数:

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

= 2 \cdot \int \frac{x}{( 2 - x ) ^{6}} d x

使用换元积分法

= 2 \cdot \int - \frac{- u + 2}{u ^{6}} d u

提出常数:

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

= 2 (- \int \frac{- u + 2}{u ^{6}} d u)

乘开

\frac{- u + 2}{u ^{6}} : - \frac{1}{u ^{5}} + \frac{2}{u ^{6}}

= 2 (- \int - \frac{1}{u ^{5}} + \frac{2}{u ^{6}} 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

= 2 (- (- \int \frac{1}{u ^{5}} d u + \int \frac{2}{u ^{6}} d u))

\int \frac{1}{u ^{5}} d u = - \frac{1}{4 u ^{4}}

\int \frac{2}{u ^{6}} d u = - \frac{2}{5 u ^{5}}

= 2 (- (- (- \frac{1}{4 u ^{4}}) - \frac{2}{5 u ^{5}}))

u = 2 - x

代回

= 2 (- (- (- \frac{1}{4 ( 2 - x ) ^{4}}) - \frac{2}{5 ( 2 - x ) ^{5}}))

化简

= - 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}})

解答补常数

= - 2 (\frac{1}{4 ( 2 - x ) ^{4}} - \frac{2}{5 ( 2 - x ) ^{5}}) + C

作图

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