∫(pai,0) sin^3(x/2) dx 的定积分是?

题目

∫(pai,0) sin^3(x/2) dx 的定积分是?

答案

∫[0,π]sin^3(x/2) dx
=-2∫[0,π]sin^2(x/2) dcos(x/2)
= -2∫[0,π][1-cos^2(x/2) ]dcos(x/2)
=[-2cos(x/2)+2/3cos^3(x/2)][0,π]
=2-2/3
=4/3