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受欢迎的微积分 >

integral of 1/(sqrt((x^2-1)^3))

解答

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

解答

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

求解步骤

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

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

假定

(x^{2} - 1) \geq 0

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

使用三角换元法

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

化简

cot^{2} (u) sec (u) : cot (u) csc (u)

= \int cot (u) csc (u) d u

使用换元积分法

= \int - 1 d v

常数积分:

\int a d x = a x

= (- 1) v

代回

= (- 1) csc (arcsec (x))

化简

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

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

解答补常数

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

作图

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x \to 1 lim (e^{\frac{1}{1 - x}})

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