∫x(arctanx)^2dx这个用分部积分法怎么求

Question

∫x(arctanx)^2dx这个用分部积分法怎么求

Answer 1

原式=(1/2)*∫arctan^xd(x^)
=(1/2)*arctan^x*x^-(1/2)*∫x^d(arctan^x)
=x^*arctan^x/2 -(1/2)*∫x^*[2*arctanx/(1+x^)]dx
=x^*arctan^x/2-∫[x^/(1+x^)]*arctanx*dx
=x^*arctan^x/2-∫arctanxdx+∫arctanxdx/(1+x^)
=x^*arctan^x/2-x*arctanx+∫xd(arctanx)+∫artanx*d(arctanx)
=x^*arctan^x/2-x*arctanx+∫[x/(1+x^)]dx+arctan^x/2
=x^*arctan^x/2-x*arctanx+arctan^x/2+∫(1/2)*(1+x^)*d(1+x^)
=x^*arctan^x/2-x*arctanx+arctan^x/2+ln(1+x^)/2