Question

∫xsin^3xdx 上限π 下限0 求定积分

Answer 1

∫x *(sinx)^3 dx
=-∫ x *(sinx)^2 d(cosx)
= ∫ x *(cosx)^2 -x d(cosx)
而显然
∫ x *(cosx)^2 d(cosx)
=1/3 *∫ x d(cosx)^3
= x/3 *(cosx)^3 -∫1/3 *(cosx)^3dx
= x/3 *(cosx)^3 -∫1/3 *(cosx)^2 d(sinx)
= x/3 *(cosx)^3 -∫1/3 -(sinx)^2 /3 d(sinx)
= x/3 *(cosx)^3 -1/3 *sinx +1/9 *(sinx)^3
∫-x d(cosx)
= -x *cosx +∫cosx dx
= -x *cosx +sinx
二者相加得到
∫x *(sinx)^3 dx
= x/3 *(cosx)^3 +2/3 *sinx +1/9 *(sinx)^3 -x *cosx
代入上下限π和0，
定积分值= -π/3 +π= 2π/3