integral of csc^9(x)

解答

\int csc^{9} (x) d x

- \frac{cos ( x ) csc ^{8} ( x )}{8} + \frac{7}{8} (- \frac{1}{6} csc^{6} (x) cos (x) + \frac{5}{48} (- 2 csc^{4} (x) cos (x) + 3 (- cot (x) csc (x) + ln tan (\frac{x}{2})))) + C

求解步骤

\int csc^{9} (x) d x

使用降次积分法

= - \frac{cos ( x ) csc ^{8} ( x )}{8} + \frac{7}{8} \cdot \int csc^{7} (x) d x

\int csc^{7} (x) d x = - \frac{1}{6} csc^{6} (x) cos (x) + \frac{5}{48} (- 2 csc^{4} (x) cos (x) + 3 (- cot (x) csc (x) + ln tan (\frac{x}{2})))

= - \frac{cos ( x ) csc ^{8} ( x )}{8} + \frac{7}{8} (- \frac{1}{6} csc^{6} (x) cos (x) + \frac{5}{48} (- 2 csc^{4} (x) cos (x) + 3 (- cot (x) csc (x) + ln tan (\frac{x}{2}))))

解答补常数

= - \frac{cos ( x ) csc ^{8} ( x )}{8} + \frac{7}{8} (- \frac{1}{6} csc^{6} (x) cos (x) + \frac{5}{48} (- 2 csc^{4} (x) cos (x) + 3 (- cot (x) csc (x) + ln tan (\frac{x}{2})))) + C