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∫xsinnxDx

函数y=xsinnx的原函数是∫xsinnxdx若n=0,∫xsinnxdx=C若n≠0,∫xsinnxdx=(-1/n)∫xd(cosnx)=(-1/n)xcosnx+(1/n)∫cosnxdx=(-1/n)xcosnx+(1/n^2)sinnx+C希望能帮到你!

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∫[0,π]x*sin nx dx=-1/n∫[0,π]x dcos nx=-1/nxcosnx[0,π]+1/n∫[0,π]cos nx dx=-1/nxcosnx[0,π]=-1/nπ (n是偶数时)=1/nπ (n是奇数时)

原式=-1/n∫(0,π)xdcosnx=-1/nxcosnx|(0,π)+1/n∫(0,π)cosnxdx=-1/n πcosnπ+0+1/n平方sinnx|(0,π)=-πcosnπ1/n

e^xsin(nx)=(e^{x+inx}-e^{x-inx})/2.然后就变成了形如e^{cx}dx的积分

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

∫(0→π/2) e^xsinx dx = - ∫(0→π/2) e^x dcosx= - e^xcosx:(0→π/2) + ∫(0→π/2) e^xcosx dx,分部积分法= 1 + ∫(0→π/2) e^x dsinx= 1 + e^xsinx:(0→π/2) - ∫(0→π/2) e^xsinx dx2∫(0→π/2) e^xsinx dx = 1 + e^(π/2)==> ∫(0→π/2) e^xsinx dx = [1 + e^(π/2)]/2∫(-π/

x=e^t ∫lnxdx=∫t^2de^t=t^2*e^t-∫e^tdt^2=t^2*e^t-2∫t*e^tdt=t^2*e^t-2∫tde^t=t^2*e^t-2(t*e^t-∫e^tdt)=t^2*e^t-2(t*e^t-e^t)=e^t(t^2-2t+2) 代入t=inx x(lnx-2inx+2)

=xarctanx-∫xdarctanx=xarctanx-∫x/(1+x)dx=xarctanx-1/2∫1/(1+x)d(1+x)=xarctanx-1/2ln(1+x)+C

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