積分上是變?cè)?br/>先拆分
∫x[f(x)+f(-x)]dx
=∫[-a,0]xf(x)dx+∫[0,a]xf(x)dx+∫[-a,0]xf(-x)dx+∫[0,a]xf(-x)dx
對(duì)于第三第四個(gè)進(jìn)行變?cè)獃=-x,注意上下限也變
=∫[-a,0]xf(x)dx+∫[0,a]xf(x)dx+∫[a,0]-yf(y)(-dy)+∫[0,-a]-yf(y)(-dy)
=∫[-a,0]xf(x)dx+∫[0,a]xf(x)dx+∫[a,0]yf(y)dy+∫[0,-a]yf(y)dy
再另x=y
=∫[-a,0]xf(x)dx+∫[0,-a]xf(x)dx+∫[a,0]xf(x)dx+∫[0,a]xf(x)dx
上下限交換,多出個(gè)負(fù)號(hào)
=∫[-a,0]xf(x)dx-∫[-a,0]xf(x)dx+∫[a,0]xf(x)dx-∫[a,0]xf(x)dx
=0