如果答案是(1/6)x^3+(1/2)(x^2)sinx+xcosx-sinx+C,那么你的題目抄錯咯,題目應(yīng)該是：
∫x²cos²(x/2) dx,那么
∫ x²cos²(x/2)dx
=∫1/2x²(1+cosx)dx
=1/2∫ x²dx+∫1/2x²cosxdx
=1/6 x³+1/2∫x²dsinx
=1/6 x³+1/2 x²sinx-∫xsinxdx
=1/6 x³+1/2 x²sinx-[-xcosx+∫cosxdx]
=1/6 x³+1/2 x²sinx+ xcosx-sinx+C
如果題目是∫ x²cos(x/2)dx,那么,參考答案錯了,正確的答案是：
∫ x²cos(x/2)dx=∫2x²dsin(x/2)
= 2x²sin(x/2)- ∫2 sin(x/2) dx²
=2x²sin(x/2)- ∫4xsin(x/2) dx
=2x²sin(x/2)- ∫-8x dcos(x/2)
=2x²sin(x/2)+8xcos(x/2)- ∫8cos(x/2)dx
=2x²sin(x/2)+8xcos(x/2)- 16sin(x/2)+C