1/(sinx)^m ,不定积分

题目

1/(sinx)^m ,不定积分

答案

最厚的积分书都已经查了,非整数不存在公式!
整数m 则是逐一可解的:
m=1,-(log(cos(x)+1)-log(cos(x)-1))/2
m=2,-1/tan(x)
m=3,-((cos(x)^2-1)*log(cos(x)+1)+(1-cos(x)^2)*log(cos(x)-1)-2*cos(x))/(4*cos(x)^2-4)
m=4,-(3*tan(x)^2+1)/(3*tan(x)^3).
m=5,-(3*sin(x)^4*log(cos(x)+1)-3*sin(x)^4*log(cos(x)-1)+6*cos(x)*sin(x)^2+4*cos(x))/(16*sin(x)^4)
m=6,-(8*cos(x)^5-20*cos(x)^3+15*cos(x))/(15*sin(x)^5)
m=7,-(15*sin(x)^6*log(cos(x)+1)-15*sin(x)^6*log(cos(x)-1)+30*cos(x)*sin(x)^4+20*cos(x)*sin(x)^2+16*cos(x))/(96*sin(x)^6)
m=8,(16*cos(x)^7-56*cos(x)^5+70*cos(x)^3-35*cos(x))/(35*sin(x)^7)
m=9,-(105*sin(x)^8*log(cos(x)+1)-105*sin(x)^8*log(cos(x)-1)+210*cos(x)*sin(x)^6+140*cos(x)*
sin(x)^4+112*cos(x)*sin(x)^2+96*cos(x))/(768*sin(x)^8)
m=10,-(128*cos(x)^9-576*cos(x)^7+1008*cos(x)^5-840*cos(x)^3+315*cos(x))/(315*sin(x)^9)
...
但也不存在简易的一般公式!