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Cosx的四次方的微积分

y=(cosx)^4 dy = 4(cosx)^3.(-sinx) dx dy = -4(cosx)^3.sinx dx

∫(cosx)^3dx=∫(cosx)^2dsinx=∫[1-(sinx)^2]dsinx=∫dsinx-∫(sinx)^2dsinx=sinx-(1/3)(sinx)^3+c 望采纳,如果不妥请回复.

∫(cosx)^4 dx=∫(1-sinx^2)cosx^2dx=∫cosx^2dx-∫sinx^2cosx^2dx=∫(1/2)(1+cos2x)x-∫(1/4)[(1-cos4x)/2]dx=(x/2)+(1/4)sin2x-(x/8)+(1/32)sin4x+C =3x/8+(1/4)sin2x+(1/32)sin4x+C

cosx的4次方的从0到π因为cosx的4次方是以π 为周期的函数所以∫(0,π)cosx的4次方dx=∫(-π/2,π/2)cosx的4次方dx=2(0,π/2)cosx的4次方dx=2*3/4*1/2*π/2=3π/8

∫(cosx)^4dx=∫[(1+cos2x)/2]^2dx=1/4∫[1+2cos2x+(cos2x)^2]dx=1/4∫dx+1/4∫2cos2xdx+1/4∫(cos2x)^2dx=x/4+C+1/4∫cos2xd(2x)+1/4∫[(1+cos4x)/2]dx=x/4+(sin2x)/4+C+1/4∫1/2dx+1/4∫(cos4x)/2dx=3x/8+(sin2x)/4+C+1/32∫4cos4xdx=3x/8+(sin2x)/4+C+1/32∫cos4xd(4x)=3x/8+(sin2x)/4+(sin4x)/32+C

=(cosx^2)^2 =(0.5cos2x+0.5)^2 =0.25cos2x^2+0.5cos2x+0.25 =0.125cos4x+0.125+0.5cos2x+0.25 =0.125cos4x+0.5cos2x+0.375 第一项:∫0.125/4*dsin4x 第二项:∫0.5/2*dsin2x 原式=sin4x/16+sin2x/4+3/8+C =sin4x/16+sin2x/4+C

降幂:∫(cosX)^4 dx=∫(1+cos2X)/2 dx=(1/2)*∫[1+(1+cos4X)/2]dx=(1/2)*[∫(3/2)dx + ∫cos4X/2 dx] =(1/2)*[(3/2)X+sin4X/8]+C=(3/4)X+sin4X/16 +C

cos^4 x=(cos^2 x)^2=[(1+cos2x)/2]^2=(1/4)[1+2cos2x+cos^2 2x]=(1/4)[1+2cos2x+(1+cos4x)/2]=(1/8)[3+4cos2x+cos4x]积分=(1/8)[3x+2sin2x+(1/4)sin4x]=(1/32)[12x+8sin2x+sin4x]

原式=(1/4)∫(1+cos2x)^2dx=(1/4)∫[1+2cos2x+(cos2x)^2]dx=x/4+(sinx)/4+(1/8)∫(1+cos4x)dx=x/4+(sinx)/4+x/8+(sin4x)/32+C=3x/8+(sinx)/4+(sin4x)/32+C

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