sin^8x在派到0的定积分

Question

sin^8x在派到0的定积分

Answer 1

∫(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