求不定积分
1.∫xtan²xdx
=∫x（sec²x-1）dx
=∫xsec²xdx-∫xdx
=xtanx-∫tanxdx-∫xdx
=xtanx+㏑|cosx|-x²/2+C
2.∫xln(x - 1) dx
= ∫ln(x - 1) d(x²/2)
= (1/2)x²ln(x - 1) - (1/2)∫x² dln(x - 1)
= (1/2)x²ln(x - 1) - x²/4 - (1/2)x - (1/2)ln|x - 1| + C
3.∫e^x(sinx)^2dx
=1/2∫e^x(1-cos2x)dx
=1/2∫(e^x-e^xcos2x)dx
=1/2∫e^xdx-1/2∫e^xcos2xdx
=1/2e^x-1/2∫e^xcos2xdx
4.∫(x^2-1)sin2xdx
=∫x^2*sin2xdx-∫sin2xdx
=-1/2*∫x^2dcos2x-1/2*∫sin2xd2x
=1/2*∫2xcos2xdx-1/2*x^2*cos2x+1/2*cos2x+C
=1/2*xsin2x+1/4*cos2x-1/2*x^2*cos2x+1/2*cos2x+C

5.∫e^√x dx
令√x=t
x=t^2
dx=2tdt
则原式化为=∫e^t*2tdt
=2∫tde^t
=2te^t-2∫e^tdt
=2e^√x*(√x-1)+C
6.∫x^2arctanxdx
= 1/3 ∫ arctanx dx³
= x³/3 arctanx - 1/3 ∫ x³/(1+x²) dx
= x³/3 arctanx - 1/3 ∫ [x - x/(1+x²)] dx
= x³/3 arctanx - x²/6 - 1/6 ln(1+x²) +C
7∫x(cscx)^2dx
=∫x/(sinx)^2dx
=1/2∫sint/(sint)^2dt
=-1/2∫1/(1-(cost)^2)d(cost)
=-1/4(∫[1/(1-cost)+1/(1+cost)]d(cost))
= -1/4ln[(1+cos(x^2))/(1-cos(x^2))]+C

看的头疼。。。

nice

头晕
　　——切，前排神马的，我、我才不会在乎呢~