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受欢迎的微积分 >

integral of (10)/(((ax)^2-b^2)^{3/2)}

解答

\int \frac{10}{( ( a x ) ^{2} - b ^{2} ) ^{\frac{3}{2}}} d x

解答

- \frac{10 x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{10}{( ( a x ) ^{2} - b ^{2} ) ^{\frac{3}{2}}} d x

提出常数:

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

= 10 \cdot \int \frac{1}{( ( a x ) ^{2} - b ^{2} ) ^{\frac{3}{2}}} d x

使用换元积分法

= 10 \cdot \int \frac{1}{a ( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

提出常数:

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

= 10 \cdot \frac{1}{a} \cdot \int \frac{1}{( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

使用三角换元法

= 10 \cdot \frac{1}{a} \cdot \int \frac{sec ( v )}{b ^{2} tan ^{2} ( v )} d v

提出常数:

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

= 10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{sec ( v )}{tan ^{2} ( v )} d v

使用三角恒等式改写

= 10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{cos ( v )}{sin ^{2} ( v )} d v

使用换元积分法

= 10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{1}{w ^{2}} d w

使用幂法则

= 10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{w})

代回

= 10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{b} a x ) )})

化简

10 \cdot \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{b} a x ) )}) : - \frac{10 x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}}

= - \frac{10 x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}}

解答补常数

= - \frac{10 x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

作图

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