∫x^2arctanxdx
=(1/3)∫arctanxd(x^3)
=(1/3)x^3.arctanx -(1/3)∫x^3/(1+x^2) dx
consider
x^3 = x(1+x^2) -x
=x(1+x^2) -(1/2)(2x)
∫x^2arctanxdx
=(1/3)x^3.arctanx -(1/3)∫x^3/(1+x^2) dx
=(1/3)x^3.arctanx -(1/3)∫xdx + (1/6)∫2x/(1+x^2) dx
=(1/3)x^3.arctanx -(1/6)x^2 + (1/6)ln|1+x^2| + C