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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用三角换元法

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

= \frac{1}{2} \cdot \int \frac{1}{v ^{2}} d v

使用幂法则

= \frac{1}{2} (- \frac{1}{v})

代回

= \frac{1}{2} - \frac{1}{sin ( arcsec ( \frac{1}{2 ^{\frac{1}{2}}} x ) )}

化简

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

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

解答补常数

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

作图

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