pure reduction formula(urgent)

F(m+2,n+2)=∫[0,a] (sinx)^(m+2)/(cosx)^(n+2) dx (a=pi/4)
=∫[0,a] (sinx)^m/(cosx)^n *(secx)^2 dx-∫[0,a] (sinx)^m/(cosx)^n dx
(by parts)=(sinx)^m/(cosx)^n*tanx|[0~a]
-∫[0,a] [(m+1)(sinx)^m/(cosx)^n+n(sinx)^(m+2)/(cosx)^(n+2)] dx
=(1/√2)^(m-n)-(m+1)F(m,n)-nF(m+2,n+2)
then
F(m+2,n+2)=1/(n+1)*(1/√2)^(m-n)-(m+1)/(n+1)F(m,n)

note: (by parts) u=(sinx)^m/(cosx)^n, dv=(secx)^2 dx


圖片參考：http://imgcld.yimg.com/8/n/AD04686329/o/701006190171213873424510.jpg