定积分（1-cost）^（2／5）dt.t的范围是从0到2派

Question

定积分（1-cost）^（2／5）dt.t的范围是从0到2派

Answer 1

∫[0,2π](1-cost)^(5/2) dt
=∫[0,2π][2sin(t/2)^2]^(5/2)dt
=8√2∫[0,2π]sin(t/2)^5 d(t/2)
=-8√2∫[0,2π]sin(t/2)^4d(cos(t/2))
=-8√2∫[0,2π](1-cos(t/2)^2)^2dcos(t/2)
=-8√2∫[0,2π] [1-2cos(t/2)+cos(t/2)^4 ] dcos(t/2)
=-8√2[(cosπ-(cosπ)^2+(1/5)(cosπ)^5 )-(cos0-cos0^2+(1/5)cos0^5)]
=-8√2(-2-1/3-1/5)
=(144√2) /15