(1/2)∫dx/(tan(x/2)cos^2(x/2))

题目

(1/2)∫dx/(tan(x/2)cos^2(x/2))
求这个的积分,
还是不是很懂，为什么将tanx/2视为中间变量,其导数刚好是1/2cos^2(x/2))

答案

(1/2)∫dx/(tan(x/2)cos^2(x/2))
=∫d(1/2x)/(tan(x/2)cos^2(x/2))
设x=x/2
原式=∫dx/(tanxcos^2x)
=∫dx/((sinx/cosx)cos^2x)
=∫dx/(sinxcosx)
=∫(sin^2x+cos^2x)dx/(sinxcosx)
=∫(sinx/cosx)dx+∫(cosx/sinx)dx+C
=∫d(cosx)/cosx-∫d(sinx)/sinx+C
=ln(cosx/sinx)+C=lncotx +C