数学题(用不定积分做,前面的符号我就不写了) 1.(e^x/1+e^2x)dx 2.(lnx/根

1.原式=∫1/1+e^2x)d(e^x )=arctan(e^x )+C
2.原式=∫lnx/√xdx=2∫lnxd(√x)=2lnx*√x-2∫√x*1/xdx=2lnx*√x-2∫1/√xdx
=2lnx*√x-4∫d(√x)
=2lnx*√x-4√x+C
3.∫arcsin2xdx=x*arcsin2x-∫xd(arcsin2x)=x*arcsin2x-2∫x/(1-4x^2)dx
=x*arcsin2x-1/2∫(1/(1-2x)-1/(1+2x))dx
=x*arcsin2x-1/2∫(1/(1-2x))dx+1/2∫(1/(1+2x))dx
=x*arcsin2x+1/4*ln(1-2x)+1/4*ln(1+2x)+C
=x*arcsin2x+1/4*ln(1-4x^2)+C
4.∫x^5lnxdx=1/6*∫lnxd(x^6) =1/6*x^6*lnx-1/6∫x^6*1/xdx =1/6*x^6*lnx-1/6∫x^5dx
=1/6*x^6*lnx-1/36∫d(x^6)
=1/6*x^6*lnx-1/36*x^6+C