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受欢迎的微积分 >

integral of (11w+4)/((3+6w-w^2)^2)

解答

\int \frac{11 w + 4}{( 3 + 6 w - w ^{2} ) ^{2}} d w

解答

\frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )}) + C

求解步骤

\int \frac{11 w + 4}{( 3 + 6 w - w ^{2} ) ^{2}} d w

对

3 + 6 w - w^{2}

配方:

- (w - 3)^{2} + 12

= \int \frac{11 w + 4}{( - ( w - 3 ) ^{2} + 12 ) ^{2}} d w

使用换元积分法

= \int \frac{11 u + 37}{( - u ^{2} + 12 ) ^{2}} d u

乘开

\frac{11 u + 37}{( - u ^{2} + 12 ) ^{2}} : \frac{11 u}{( - u ^{2} + 12 ) ^{2}} + \frac{37}{( - u ^{2} + 12 ) ^{2}}

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{11 u}{( - u ^{2} + 12 ) ^{2}} d u + \int \frac{37}{( - u ^{2} + 12 ) ^{2}} d u

\int \frac{11 u}{( - u ^{2} + 12 ) ^{2}} d u = \frac{11}{2 ( - u ^{2} + 12 )}

\int \frac{37}{( - u ^{2} + 12 ) ^{2}} d u = 37 (\frac{1}{96 3} ln u + 23 - \frac{1}{48 ( u + 2 3 )} - \frac{1}{96 3} ln u - 23 - \frac{1}{48 ( u - 2 3 )})

= \frac{11}{2 ( - u ^{2} + 12 )} + 37 (\frac{1}{96 3} ln u + 23 - \frac{1}{48 ( u + 2 3 )} - \frac{1}{96 3} ln u - 23 - \frac{1}{48 ( u - 2 3 )})

u = w - 3

代回

= \frac{11}{2 ( - ( w - 3 ) ^{2} + 12 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )})

乘开

- (w - 3)^{2} + 12 : - w^{2} + 6 w + 3

= \frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )})

解答补常数

= \frac{11}{2 ( - w ^{2} + 6 w + 3 )} + 37 (\frac{1}{96 3} ln w - 3 + 23 - \frac{1}{48 ( w - 3 + 2 3 )} - \frac{1}{96 3} ln w - 3 - 23 - \frac{1}{48 ( w - 3 - 2 3 )}) + C

作图

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