Question

求定积分∫（0,π） √（sin^3 x-sin^5 x)dx

Answer 1

I=∫√（sin^3 x-sin^5 x)dx (0,π)

= ∫ sinx√(sinx-(sinx)^2 ) dx

let

a = sinx

da = cosx dx

x=0,a=0

x=π,a =1

I= ∫ a √ (a- a^2) da/( √(1-a^2) (0,1)

= ∫ a^(3/2) da (0,1)

= (2/5)[a^(5/2)] (0,1)

=2/5