∫xcos^3 xdx

Question

这个貌似要用分部积分法做 我做了一半 僵住了 做不下去了~ 求高人~

Answer 1

∫xcos^3 xdx
=∫xcos^2 xcosxdx
=∫xcos^2 xdsinx
=∫x(1-sin^2 x)dsinx
=∫xdsinx-∫xsin^2 x dsinx
=xsinx-∫sinxdx-1/3∫xdsin^3 x
=xsinx+∫dcosx-1/3(xsin^3 x -∫sin^3 x dx)
=xsinx+cosx-xsin^3 x/3+1/3∫sin^2xsinxdx
=xsinx+cosx-xsin^3 x/3-1/3∫(1-cos^2 x dcosx)
=xsinx+cosx-xsin^3 x/3-1/3∫dcosx+1/3∫cos^2 x dcosx
=xsinx+cosx-xsin^3 x/3-cosx/3+cos^3x/6+C
不知道答案对不对。不对自已按这个方法算。