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∫xsin4xDx

∫xsin4xdx=(-1/4)∫xdcos4x=(-1/4)xcos4x+(1/4)∫cos4xdx=(-1/4)xcos4x+(1/16)sin4x+C

∫xcos4xdx=xsin4x/4-∫sin4x/4dx=xsin4x/4+cos4x/16+C 如果不懂,请追问,祝学习愉快!

∫xsinxdx=-∫xdcosx=-xcosx+∫cosxdx=-xcosx+sinx+c

∫x(sinx)^4dx=(1/4)∫x(1-cos2x)^2dx=(1/4)∫[x-2xcos2x +x(cos2x)^2 ]dx=(1/8)∫[2x-4xcos2x +x(1+cos4x) ]dx=(1/8)∫[3x-4xcos2x +xcos4x ]dx=(1/8)[(3/2)x^2-∫ 4xcos2xdx +∫xcos4x dx ] consider ∫4xcos2xdx=2∫xdsin2x=2xsin2x - 2∫sin2x dx=2xsin2x +cos2x

∫xsinxdx =Sx*(cos2x+1)/2 dx=1/2*Sxcos2xdx+1/2*Sxdx=1/4*Sxdcos2x+1/4*x^2=1/4*xcos2x-1/4*Scos2xdx+1/4*x^2=1/4*xcos2x-1/8*sin2x+1/4*x^2+c

原式=-∫xd(cosx) =-xcosx+∫cosxdx (分部积分法) =-xcosx+sinx+C (C是积分常数).

∫xsin^2xdx=1/2∫x(1-cos2x)dx=1/4x^2-1/2∫xcos2xdx=1/4x^2-1/4∫xdsin2x=1/4x^2-1/4xsin2x+1/4∫sin2xdx=1/4x^2-1/4xsin2x-1/8cos2x+c

用分部积分法:∫xsinxdx = -∫x(-sinx)dx = -∫xdcosx= -(xcosx - ∫cosxdx)= -(xcosx - sinx) + C= sinx - xcosx + C

∫ e^xsinx dx=(1/2)∫ e^x(1-cos2x) dx=(1/2)e^x - (1/2)∫ e^xcos2x dx (1) 下面计算:∫ e^xcos2x dx=∫ cos2x d(e^x) 分部积分=e^xcos2x + 2∫ e^xsin2x dx=e^xcos2x + 2∫ sin2x d(e^x) 再分部积分=e^xcos2x + 2e^xsin2x - 4∫ e^xcos2x dx 将 -4∫ e^xcos2x

解:原式=-∫xd(cosx) =-xcosx+∫cosxdx (应用分部积分法) =-xcosx+sinx+C (C是积分常数).

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