求 ∫ x^3 • cos³x dx

用分部积分法原式=∫ x^3 • (cosx)^2 • cosxdx=∫ x^3 • [1-(sinx)^2] d(sinx)=∫ x^3d(sinx)-∫ x^3 • (sinx)^2d(sinx)因∫ x^3d(sinx)=x^3 •sinx-∫ sinxd(x^3)=x^3 •sinx-3∫ x^2 •sinxdx=x^3 •sinx+3∫ x^2 d(cosx)=x^3 •sinx+3[x^2•cosx-∫ cosxd(x^2)]=x^3 •sinx+3(x^2•cosx-2∫ x •cosxdx)=x^3 •sinx+3x^2•cosx-6∫ x •d(sinx)=x^3 •sinx+3x^2•cosx-6(x •sinx-∫ sinxdx)=x^3 •sinx+3x^2•cosx-6x •sinx+6∫ sinxdx=x^3 •sinx+3x^2•cosx-6x •sinx-6cosx+C1又∫ x^3 • (sinx)^2d(sinx)（I）=(1/3)∫ x^3d[(sinx)^3]=(1/3)[x^3 •(sinx)^3-∫ (sinx)^3d(x^3)]=(1/3)x^3 •(sinx)^3-(1/3)∫ (sinx)^3d(x^3)=(1/3)x^3 •(sinx)^3-∫ x^2•(sinx)^3dx而∫ x^2•(sinx)^3dx（II）=∫ x^2•[(cosx)^2-1]d(cosx)=∫ x^2•(cosx)^2d(cosx)-∫ x^2d(cosx)其中∫ x^2d(cosx)=x^2•cosx-∫ cosxd(x^2)=x^2•cosx-2∫ x•cosxdx=x^2•cosx-2∫ xd(sinx)=x^2•cosx-2(x•sinx-∫ sinxdx)=x^2•cosx-2x•sinx+2∫ sinxdx=x^2•cosx-2x•sinx-2cosx+C2其中∫ x^2•(cosx)^2d(cosx)（III）=(1/3)∫ x^2d[(cosx)^3]=(1/3)[x^2•(cosx)^3-∫ (cosx)^3d(x^2)]=(1/3)x^2•(cosx)^3-(2/3)∫ x•(cosx)^3dx而∫ x•(cosx)^3dx=∫ x• [1-(sinx)^2] d(sinx)=∫ xd(sinx)-∫ x• (sinx)^2d(sinx)其中∫ xd(sinx)=x• sinx-∫ sinxdx=x• sinx+cosx+C3其中∫ x• (sinx)^2d(sinx)=(1/3)∫ xd [(sinx)^3]=(1/3)[x•(sinx)^3-∫ (sinx)^3dx]=(1/3)x•(sinx)^3-(1/3)∫ [(cosx)^2-1]d(cosx)=(1/3)x•(sinx)^3-(1/3)[∫ (cosx)^2d(cosx)-∫ d(cosx)]=(1/3)x•(sinx)^3-(1/3)[(1/3)(cosx)^3-cosx]+C4=(1/3)x•(sinx)^3-(1/9)(cosx)^3+(1/3)cosx+C4则∫ x•(cosx)^3dx=(x• sinx+cosx+C3)-[(1/3)x•(sinx)^3-(1/9)(cosx)^3+(1/3)cosx+C4]=x• sinx+(2/3)cosx-(1/3)x•(sinx)^3+(1/9)(cosx)^3+C5于是（III）式∫ x^2•(cosx)^2d(cosx)=(1/3)x^2•(cosx)^3-(2/3)∫ x•(cosx)^3dx=(1/3)x^2•(cosx)^3-(2/3)[x• sinx+(2/3)cosx-(1/3)x•(sinx)^3+(1/9)(cosx)^3+C5]=(1/3)x^2•(cosx)^3+(2/3)x•(sinx)^3-(2/3)x• sinx-(2/27)(cosx)^3-(4/9)cosx+C6于是（II）式∫ x^2•(sinx)^3dx=[(1/3)x^2•(cosx)^3+(2/3)x•(sinx)^3-(2/3)x• sinx-(2/27)(cosx)^3-(4/9)cosx+C6]-[x^2•cosx-2x•sinx-2cosx+C2]=(1/3)x^2•(cosx)^3-x^2•cosx+(2/3)x•(sinx)^3+(4/3)x• sinx-(2/27)(cosx)^3+(14/9)cosx+C7于是（I）式∫ x^3 • (sinx)^2d(sinx)=(1/3)x^3 •(sinx)^3-(1/3)x^2•(cosx)^3+x^2•cosx-(2/3)x•(sinx)^3-(4/3)x• sinx+(2/27)(cosx)^3-(14/9)cosx+C8于是原式∫ x^3 • cos³x dx=x^3 •sinx+3x^2•cosx-6x •sinx-6cosx-(1/3)x^3 •(sinx)^3+(1/3)x^2•(cosx)^3-x^2•cosx+(2/3)x•(sinx)^3+(4/3)x• sinx-(2/27)(cosx)^3+(14/9)cosx+C=-(1/3)x^3 •(sinx)^3+x^3 •sinx+(1/3)x^2•(cosx)^3+2x^2•cosx+(2/3)x•(sinx)^3-(14/3)x •sinx-(2/27)(cosx)^3-(40/9)cosx+C