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受欢迎的微积分 >

integral of 1/((x^2-6x)^{3/2)}

解答

\int \frac{1}{( x ^{2} - 6 x ) ^{\frac{3}{2}}} d x

解答

- \frac{x - 3}{9 ( x ^{2} - 6 x ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{1}{( x ^{2} - 6 x ) ^{\frac{3}{2}}} d x

对

x^{2} - 6 x

配方:

(x - 3)^{2} - 9

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

使用换元积分法

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

使用三角换元法

= \int \frac{sec ( v )}{9 tan ^{2} ( v )} d v

提出常数:

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

= \frac{1}{9} \cdot \int \frac{sec ( v )}{tan ^{2} ( v )} d v

使用三角恒等式改写

= \frac{1}{9} \cdot \int \frac{cos ( v )}{sin ^{2} ( v )} d v

使用换元积分法

= \frac{1}{9} \cdot \int \frac{1}{w ^{2}} d w

使用幂法则

= \frac{1}{9} (- \frac{1}{w})

代回

= \frac{1}{9} (- \frac{1}{sin ( arcsec ( \frac{1}{3} ( x - 3 ) ) )})

化简

\frac{1}{9} (- \frac{1}{sin ( arcsec ( \frac{1}{3} ( x - 3 ) ) )}) : - \frac{x - 3}{9 ( x ^{2} - 6 x ) ^{\frac{1}{2}}}

= - \frac{x - 3}{9 ( x ^{2} - 6 x ) ^{\frac{1}{2}}}

解答补常数

= - \frac{x - 3}{9 ( x ^{2} - 6 x ) ^{\frac{1}{2}}} + C

作图

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(6y+2t-7)dt+(8y+6t-1)dy=0,y(-1)=2

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