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受欢迎的微积分 >

integral of x/((5-4x-x^2)^{3/2)}

解答

\int \frac{x}{( 5 - 4 x - x ^{2} ) ^{\frac{3}{2}}} d x

解答

\frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{x}{( 5 - 4 x - x ^{2} ) ^{\frac{3}{2}}} d x

对

5 - 4 x - x^{2}

配方:

- (x + 2)^{2} + 9

= \int \frac{x}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{3}{2}}} d x

使用换元积分法

= \int \frac{u - 2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u

乘开

\frac{u - 2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} : \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}}

= \int \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} - \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} 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{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u - \int \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u

\int \frac{u}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u = \frac{1}{( - u ^{2} + 9 ) ^{\frac{1}{2}}}

\int \frac{2}{( - u ^{2} + 9 ) ^{\frac{3}{2}}} d u = \frac{2 u}{9 ( 9 - u ^{2} ) ^{\frac{1}{2}}}

= \frac{1}{( - u ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 u}{9 ( 9 - u ^{2} ) ^{\frac{1}{2}}}

u = x + 2

代回

= \frac{1}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( 9 - ( x + 2 ) ^{2} ) ^{\frac{1}{2}}}

化简

\frac{1}{( - ( x + 2 ) ^{2} + 9 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( 9 - ( x + 2 ) ^{2} ) ^{\frac{1}{2}}} : \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}}

= \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}}

解答补常数

= \frac{1}{( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} - \frac{2 ( x + 2 )}{9 ( - x ^{2} - 4 x + 5 ) ^{\frac{1}{2}}} + C

作图

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