f(x)=sin(x)*sin(x),f(x)为F(x)的导函数,求F(x)

Question

f(x)=sin(x) * sin(x),f(x)为F(x)的导函数,求F(x)

Answer 1

方法一：分部积分,得出循环移项
∫sin²xdx
=-∫sinxdcosx
=-sinxcosx+∫cosxdsinx
=-sinxcosx+∫cos²xdx
=-sinxcosx+∫dx-∫sin²xdx
=x-sinxcosx-∫sin²xdx
2∫sin²xdx=x-sinxcosx
∫sin²xdx=x/2-sinxcosx/2+C
F(x)=x/2-sinxcosx/2+C
方法二：二倍角公式降次
∫sin²xdx
=[∫1-cos(2x)dx]/2
=(∫dx)/2-[∫cos(2x)d(2x)]/4
=x/2-sin(2x)/4+C
=x/2-sinxcosx/2+C
F(x)=x/2-sinxcosx/2+C