integral of 1/(x^3)csc^4(1/(x^2))

解答

\int \frac{1}{x ^{3}} csc^{4} (\frac{1}{x ^{2}}) d x

\frac{1}{6} csc^{3} (\frac{1}{x ^{2}}) cos (\frac{1}{x ^{2}}) + \frac{1}{3} cot (\frac{1}{x ^{2}}) + C

求解步骤

\int \frac{1}{x ^{3}} csc^{4} (\frac{1}{x ^{2}}) d x

使用换元积分法

= \int - \frac{csc ^{4} ( u )}{2} 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 csc^{4} (u) d u

使用降次积分法

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

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

= - \frac{1}{2} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

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

代回

= - \frac{1}{2} (- \frac{cos ( \frac{1}{x ^{2}} ) csc ^{3} ( \frac{1}{x ^{2}} )}{3} + \frac{2}{3} (- cot (\frac{1}{x ^{2}})))

化简

- \frac{1}{2} (- \frac{cos ( \frac{1}{x ^{2}} ) csc ^{3} ( \frac{1}{x ^{2}} )}{3} + \frac{2}{3} (- cot (\frac{1}{x ^{2}}))) : \frac{1}{6} csc^{3} (\frac{1}{x ^{2}}) cos (\frac{1}{x ^{2}}) + \frac{1}{3} cot (\frac{1}{x ^{2}})

= \frac{1}{6} csc^{3} (\frac{1}{x ^{2}}) cos (\frac{1}{x ^{2}}) + \frac{1}{3} cot (\frac{1}{x ^{2}})

解答补常数

= \frac{1}{6} csc^{3} (\frac{1}{x ^{2}}) cos (\frac{1}{x ^{2}}) + \frac{1}{3} cot (\frac{1}{x ^{2}}) + C

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