帮忙求一个积分：(sin2x)^2*(sinx+cosx)积分区间0到pi/4

Question

帮忙求一个积分：(sin2x)^2 * (sinx + cosx) 积分区间0到pi/4

Answer 1

∫(0,π/4) (sin2x)^2 * (sinx + cosx) dx
= ∫(0,π/4) (2sinxcosx)^2 * (sinx + cosx) dx
= 4∫(0,π/4)[ (sinx)^3.(cosx)^2 + (sinx)^2(cosx)^3]dx
= 4[ -∫x:(0,π/4) (1-(cosx)^2)(cosx)^2 d(cosx) + ∫x:(0,π/4) (1-(sinx)^2)(sinx)^2 d(sinx)
= 4{ -[(cosx)^3/3 - (cosx)^5/5] (0,π/4) + [(sinx)^3/3 - (sinx)^5/5](0,π/4)}
= 4( -1/3+1/5 )
= 4 (-2/15)
=-8/15