In=∫sin^nxdx=∫sin^(n-1)xsinxdx=-∫sin^(n-1)xdcosx=-cosxsin^(n-1)x+∫cosxdsin^(n-1)x=-cosxsin^(n-1)x+(n-1)∫cosxsin^(n-2)xcosxdx=-cosxsin^(n-1)x+(n-1)∫cos²xsin^(n-2)xdx=-cosxsin^(n-1)x+(n-1)...

　　=-cosxsin^(n-1)x+(n-1)∫[sin^(n-2)x-sin^(n)x]dxIn=-cosxsin^(n-1)x+(n-1)I(n-2)-(n-1)InnIn=-cosxsin^(n-1)x+(n-1)I(n-2)即In=-1/n·cosxsin^(n-1)x+(n-1)/nI(n-2)即这个递推公式由I1=-cosx+c一直推下去即可得到最后的理论结果。