求∫x(cosx)^3dx的积分

题目

求∫x(cosx)^3dx的积分

答案

∫ x • cos³x dx
= ∫ x • (1 - sin²x) dsinx
= ∫ x dsinx - ∫ x • sin²x dsinx
= xsinx - ∫ sinx - (1/3)∫ x dsin³x
= xsinx + cosx - (1/3)xsin³x + (1/3)∫ sin³x dx
= xsinx + cosx - (1/3)xsin³x - (1/3)∫ (1 - cos²x) dcosx
= xsinx + cosx - (1/3)xsin³x - (1/3)[cosx - 1/3 • cos³x] + C
= xsinx + cosx - (1/3)xsin³x - (1/3)cosx + (1/9)cos³x + C