Evaluate integralxtan1x dxSolution utan1 dv xdx v12x2 Then a

Evaluate integral((x*tan^-1(x)) dx)

u=tan^-1 dv= xdx v=1/2(x^2) Then, applying integration by parts: ? x*arctan(x) dx = uv - ? v du = (1/2)x^2*arctan(x) - ? x^2/[2(x^2 + 1)] dx = (1/2)x^2*arctan(x) - 1/2 ? x^2/(x^2 + 1) dx = (1/2)x^2*arctan(x) - 1/2 ? [1 - 1/(x^2 + 1)] dx = (1/2)x^2*arctan(x) - 1/2 ? dx + ? 1/(x^2 + 1) dx = (1/2)x^2*arctan(x) - (1/2)x + (1/2)arctan(x) + C