设(co)sine power - 图1为非负整数. 计算(co)sine power - 图2.

    用分部积分.

    (co)sine power - 图3%20%5Ccdot%20%5Ccos%5E%7Bk-2%7D%20u%20(-%5Csin%20u)%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20%2B%20(k-1)%20%5Cint%20%5Ccos%5E%7Bk-2%7D%20u%20(%5Csin%5E2%20u)%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccos%5E%7Bk-1%7Du%20%2B%20(k-1)%20%5Cint%20%5Ccos%5E%7Bk-2%7D%20u%20(1%20-%20%5Ccos%5E2%20u)%20%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20%2B%20(k-1)%20%5Cint%20%5Ccos%5E%7Bk-2%7Du%20%5C%2C%20du%20-%20(k-1)%5Cint%20%5Ccos%5Ek%20u%20%5C%2C%20du.%0A%5Cend%7Baligned%7D%0A#card=math&code=%5Cbegin%7Baligned%7D%0A%5Cint%20%5Ccos%5Ek%20u%20%5C%2C%20du%20%26%3D%20%5Cint%20%5Ccos%5E%7Bk-1%7D%20u%5C%2C%20d%20%5Csin%20u%20%5C%5C%0A%26%3D%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20-%20%5Cint%20%5Csin%20u%20%5Ccdot%20%28k-1%29%20%5Ccdot%20%5Ccos%5E%7Bk-2%7D%20u%20%28-%5Csin%20u%29%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20%2B%20%28k-1%29%20%5Cint%20%5Ccos%5E%7Bk-2%7D%20u%20%28%5Csin%5E2%20u%29%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccos%5E%7Bk-1%7Du%20%2B%20%28k-1%29%20%5Cint%20%5Ccos%5E%7Bk-2%7D%20u%20%281%20-%20%5Ccos%5E2%20u%29%20%5C%2C%20du%20%5C%5C%0A%26%3D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20%2B%20%28k-1%29%20%5Cint%20%5Ccos%5E%7Bk-2%7Du%20%5C%2C%20du%20-%20%28k-1%29%5Cint%20%5Ccos%5Ek%20u%20%5C%2C%20du.%0A%5Cend%7Baligned%7D%0A&id=PIMQp)

    即 (co)sine power - 图4%20c%7Bk-2%7D%20%2B%20(1-k)c_k#card=math&code=c_k%20%3D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u%20%2B%20%28k-1%29%20c%7Bk-2%7D%20%2B%20%281-k%29c_k&id=Wl9zL), 化简得

    (co)sine power - 图5c%7Bk-2%7D%20%2B%20%5Cfrac%7B1%7D%7Bk%7D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u#card=math&code=c_k%20%3D%20%281-%5Cfrac%7B1%7D%7Bk%7D%29c%7Bk-2%7D%20%2B%20%5Cfrac%7B1%7D%7Bk%7D%20%5Csin%20u%20%5Ccdot%20%5Ccos%5E%7Bk-1%7D%20u&id=Sz6t1). 同理可得

    (co)sine power - 图6s%7Bk-2%7D%20-%20%5Cfrac%7B1%7D%7Bk%7D%20%5Ccos%20u%20%5Ccdot%20%5Csin%5E%7Bk-1%7D%20u#card=math&code=s_k%20%3D%20%281-%5Cfrac%7B1%7D%7Bk%7D%29s%7Bk-2%7D%20-%20%5Cfrac%7B1%7D%7Bk%7D%20%5Ccos%20u%20%5Ccdot%20%5Csin%5E%7Bk-1%7D%20u&id=X6PEU). 所以最后只要求出(co)sine power - 图7时的值即可.

    (co)sine power - 图8%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20u%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%5Csin%202u%20%2B%20C%2C%20%5C%5C%0As_2%20%26%3D%20%5Cint%20%5Csin%5E2%20u%20%5C%2C%20du%20%3D%20%5Cint%20%5Cfrac%7B1%7D%7B2%7D(1-%5Ccos%202u)%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7B2%7Du%20-%20%5Cfrac%7B1%7D%7B4%7D%20%5Csin%202u%20%2B%20C.%0A%5Cend%7Baligned%7D%0A#card=math&code=%5Cbegin%7Baligned%7D%0Ac_0%20%26%3D%20s_0%20%3D%20u%20%2B%20C%2C%20%5C%5C%0Ac_1%20%26%3D%20%5Cint%20%5Ccos%20u%20%5C%2C%20du%20%3D%20%5Csin%20u%20%2B%20C%2C%20%5C%3B%20s_1%20%3D%20%5Cint%20%5Csin%20u%20%5C%2C%20du%20%3D%20-%5Ccos%20u%20%2B%20C%2C%20%5C%5C%0Ac_2%20%26%3D%20%5Cint%20%5Ccos%5E2%20u%20%5C%2C%20du%20%3D%20%5Cint%20%5Cfrac%7B1%7D%7B2%7D%281%2B%5Ccos%202u%29%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20u%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%5Csin%202u%20%2B%20C%2C%20%5C%5C%0As_2%20%26%3D%20%5Cint%20%5Csin%5E2%20u%20%5C%2C%20du%20%3D%20%5Cint%20%5Cfrac%7B1%7D%7B2%7D%281-%5Ccos%202u%29%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7B2%7Du%20-%20%5Cfrac%7B1%7D%7B4%7D%20%5Csin%202u%20%2B%20C.%0A%5Cend%7Baligned%7D%0A&id=J1HRg)

    用(co)sine power - 图9分别记(co)sine power - 图10与(co)sine power - 图11在(co)sine power - 图12上的定积分. 作代换(co)sine power - 图13, 就知(co)sine power - 图14%3DS(k)#card=math&code=C%28k%29%3DS%28k%29&id=ASr3h), 且有

    (co)sine power - 图15%20%3D%20%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5Ek%20u%20%5C%2C%20du%20%3D%20(1-%5Cfrac%7B1%7D%7Bk%7D)C(k-2)%2C%20%5Cquad%20C(0)%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20C(1)%3D1.%20%0A#card=math&code=C%28k%29%20%3D%20%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Ccos%5Ek%20u%20%5C%2C%20du%20%3D%20%281-%5Cfrac%7B1%7D%7Bk%7D%29C%28k-2%29%2C%20%5Cquad%20C%280%29%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20C%281%29%3D1.%20%0A&id=Ej8Y2)

    依照(co)sine power - 图16的奇偶性，得到

    (co)sine power - 图17%26%3D%20%5Cfrac%7B2n-2%7D%7B2n-1%7D%20%5Ccdot%20%5Cfrac%7B2n-4%7D%7B2n-3%7D%20%5Ccdot%20%5Cldots%20%5Ccdot%20%5Cfrac%7B2%7D%7B3%7DC(1)%20%3D%20%5Cfrac%7B(2n-2)!!%7D%7B(2n-1)!!%7D%2C%5C%5C%0AC(2n)%26%3D%5Cfrac%7B(2n-1)!!%7D%7B(2n)!!%7D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cquad%20n%20%3D%201%2C%202%2C%203%2C%20%5Cldots.%20%0A%5Cend%7Baligned%7D%0A#card=math&code=%5Cbegin%7Baligned%7D%0AC%282n-1%29%26%3D%20%5Cfrac%7B2n-2%7D%7B2n-1%7D%20%5Ccdot%20%5Cfrac%7B2n-4%7D%7B2n-3%7D%20%5Ccdot%20%5Cldots%20%5Ccdot%20%5Cfrac%7B2%7D%7B3%7DC%281%29%20%3D%20%5Cfrac%7B%282n-2%29%21%21%7D%7B%282n-1%29%21%21%7D%2C%5C%5C%0AC%282n%29%26%3D%5Cfrac%7B%282n-1%29%21%21%7D%7B%282n%29%21%21%7D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cquad%20n%20%3D%201%2C%202%2C%203%2C%20%5Cldots.%20%0A%5Cend%7Baligned%7D%0A&id=fblmK)