★★★∫cos(x/2)cos(nx)dx （0→π）定积分★★★

题目

★★★∫cos(x/2)cos(nx)dx （0→π）定积分★★★

答案

利用积化和差公式
cos(x/2)cos(nx)=(1/2)[cos(n+1/2)x+cos(n-1/2)x]
积分=(1/2)∫[cos(n+1/2)x+cos(n-1/2)x]dx
=(1/(2n+1))sin[(n+1/2)x]+(1/(2n-1))sin[(n-1/2)x] |
=(1/(2n+1))sin[(n+1/2)π]+(1/(2n-1))sin[(n-1/2)π]
=(1/(2n+1))cosnπ-(1/(2n-1))cosnπ
=[(1/(2n+1))-(1/(2n-1))]cosnπ
=2*(-1)^(n+1)/(4n^2-1)