sin^8x在派到0的定积分

∫(sinx)^8dx
=-∫ (sinx)^7 dcosx
= -cosx (sinx)^7 + ∫ 7(cosx)^2(sinx)^6 dx
=-cosx (sinx)^7 +7∫ (1- (sinx)^2)(sinx)^6 dx
8∫(sinx)^8dx= -cosx (sinx)^7 + 7∫(sinx)^6dx
∫(sinx)^8dx
= (1/8) [-cosx (sinx)^7 + 7∫(sinx)^6dx]
= (1/8) {-cosx (sinx)^7 + (7/6)[-cosx (sinx)^5 + 5∫(sinx)^4dx]}
=(1/8){ -cosx (sinx)^7 + (7/6)(-cosx (sinx)^5 + (5/4)[-cosx(sinx)^3+3∫(sinx)^2dx]) }
=(1/8){ -cosx (sinx)^7 + (7/6)(-cosx (sinx)^5 + (5/4)[-cosx(sinx)^3+(3/2)[x-sin2x/2]) } + C

原函数为 1/8cos8x c.代入π和零，得结果为0.

=亅(0,pi/2)+亅(pi/2),pi) 对第二个积分代换u=pi-x代入得: 亅(pi/2,pi)dx=-亅(pi/2,0)du=亅(0,pi/2)du,所以: 亅(0,pi)sin^8xdx=2亅(0,pi/2)sin^8xdx =2(7/8)(5/6)(3/4)(1/2)(pi/2) =35pi/128