Question 401207
Applying integration by parts, if we let


{{{u = arctan x}}} --> {{{du = (1/(x^2 + 1)) dx}}}


{{{dv = x*dx}}} --> {{{v = x^2/2}}}, then


{{{int((x arctan(x)), dx) = arctan(x)(x^2/2) - int(x^2/2(x^2+1), dx)}}}


I'll rewrite {{{int(x^2/2(x^2+1), dx)}}} as {{{int((1/2) - 1/2(x^2+1), dx)}}}, which is equal to {{{int(1/2, dx) - (1/2)int(1/(x^2+1), dx)}}}.


Therefore the integral is equal to


{{{arctan(x)(x^2/2) - (  int(1/2, dx) - (1/2)int(1/(x^2+1), dx))}}}


= {{{arctan(x)(x^2/2) - x/2 + (1/2)(arctan(x)) + C}}}


= {{{(1/2)((x^2+1)(arctan(x)) - x) + C}}}, where C is a constant