求x^2 cos^3(x)的积分

题目

求x^2 cos^3(x)的积分

答案

求不定积分：∫x² cos³xdx
由于cos³x=(1/4)(3cosx+cos3x),故：
原式=∫x²[(1/4)(3cosx+cos3x)dx=(1/4)(3∫x²cosxdx+∫x²cos3xdx)
=(1/4)[3∫x²dsinx+(1/3)∫x²dsin3x]=(1/4)[3(x²sinx-2∫xsinxdx)+(1/3)(x²sin3x-2∫xsin3xdx)]
=(1/4){3(x²sinx+2∫xdcosx)+(1/3)[x²sin3x+(2/3)∫xdcos3x]}
=(1/4){3[x²sinx+2(xcosx-sinx)]+(1/3)[x²sin3x+(2/3)(xcos3x-∫cos3xdx)]}
=(3/4)x²sinx+(3/2)(xcosx-sinx)+(1/12)x²sin3x+(1/18)[xcos3x-(1/3)sin3x]+C
=(3/4)x²sinx+(3/2)(xcosx-sinx)+(1/12)x²sin3x+(1/18)xcos3x-(1/54)sin3x+C