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xCosnx的不定积分

K= ∫e^xd(cosnx)=e^xcosnx-∫(cosnx)de^x=e^xcosnx-∫e^x(cosnx)dx=e^xcosnx-1/n∫e^xd(sinnx)=e^xcosnx-1/ne^xsinnx+1/n∫(sinnx)de^x=e^xcosnx-1/ne^xsinnx+1/n∫e^x(sinnx)dx=e^xcosnx-1/ne^xsinnx-1/n^2∫e^xd(sinnx)=e^xcosnx-1/ne^xsinnx-1/n^2KK=(e^xcosnx-1/ne^xsinnx)*n^2/(n^2+1)

不停地分部积分,直到出现和原式一样的积分就可以算了.∫e^xcos(nx)dx=∫cos(nx)d(e^x)=e^xcos(nx)-∫e^x*(-n)sin(nx)dx=e^xcos(nx)+n∫sin(nx)d(e^x)=e^xcos(nx)+ne^xsin(nx)-n∫e^x*ncos(nx)dx=e^xcos(nx)+ne^xsin(nx)-n^2∫e^xcos(nx)dx 所以(n^2+1)∫e^xcos(nx)dx=e^x(cos(nx)+nsin(nx))+C 所以∫e^xcos(nx)dx=e^x(cos(nx)+nsin(nx))/(n^2+1)+C

∫xcosnxdx=1/n∫xd(sinnx)=1/n(xsinnx-∫sinnxdx)=1/n(xsinnx+1/ncosnx)=(nxsinnx+cosnx)/n^2∫xsinnx=-1/n∫xd(cosnx)=-1/n(xcosnx-∫cosnxdx)=-1/n(xcosnx-1/nsinnx)=(sinnx-nxcosnx)/n^2都是利用分步积分的方法

设In=∫exsinnxdx,则In=-∫sinnxde-x=-e-xsinnx+n∫e-xcosnxdx=-e-xsinnx-n∫cosnxde-x=-e-xsinnx-ne-xcosnx-n2∫e-xsinnxdx=-e-x(sinnx+ncosnx)-n2In∴In=sinnx+ncosnxn2+1ex+C∴∫10exsinnxdx=nco

∫x^2cosnxdx =(1/n)∫x^2cosnxdnx =(1/n)∫x^2dsinnx =(1/n)x^2sinnx-(1/n)∫sinnxdx^2=(1/n)x^2sinnx-(1/n)∫2xsinnxdx=(1/n)x^2sinnx-(2/n^2)∫xsinnxdnx=(1/n)x^2sinnx+(2/n^2)∫xdcosnx=(1/n)x^2sinnx+(2/n^2)xcosnx-(2/n^2)∫cosnxdx=(1/n)x^2sinnx+(2/n^2)xcosnx-(2/n^3)∫cosnxdnx=(1/n)x^2sinnx+(2/n^2)xcosnx-(2/n^3)sinnx+C

=(3xx+1)sinnx/n-(积分号)6xsinnx/n=(3xx+1)sinnx/n+6xcosnx/(nn)-(积分号)6cosnx/(nn)=sinnx*((3xx+1)/n-6/nnn)+cosnx*6x/nn+c;

1、 ∫(sinxcosx)^2/((sinx)^3+(cosx)^3)^2 dx=∫(secx)^2(tanx)^2/((tanx)^3+1)^2 dx=∫(tanx)^2/((tanx)^3+1)^2 dtanx=1/3∫ 1/((tanx)^3+1)^2 d(tanx)^3= -1/3[(tanx)^3+1)] + C2、 (sinx)^2*cosnx=1/2{(1-cos2x)cosnx}=1/2{ cosnx - cos2xcosnx }=1/2{ cosnx -

先化作2倍角(1+cos2x)/2,再对2x积分

∫udv=uv-∫udv 多次使用分部积分,把x^2降次就行了.∫x^2.cosnx dx=1/n*∫x^2 * d(sinnx)=1/n*(x^2*sinnx-∫sinnxd(x^2))=1/n*(x^2*sinnx-∫2xsinnxdx)=1/n(x^2*sinnx+2/n*∫xd(cosnx))=x^2/n*sinnx+2/n^2*(xcosnx-∫cosnxdx)=x^2/n*sinnx+2/n^2*xcosnx-2/

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