∫ xlnx/(1+x^2)^2 dx
= 1/2*∫ lnx/(1+x^2)^2 d(1+x^2)
= -1/2*∫ lnx d[1/(1+x^2)]
= -1/2*lnx*1/(1+x^2) + 1/2*∫ [1/(1+x^2)] dlnx
= -1/2*lnx*1/(1+x^2) + 1/2*∫ x/[x^2(1+x^2)] dx
= -1/2*lnx*1/(1+x^2) + 1/4*∫ [1/x^2 - 1/(1+x^2)] dx^2
= -1/2*lnx*1/(1+x^2) + 1/4*ln(x^2) - 1/4*ln(1+x^2) + C
= -1/2*lnx*1/(1+x^2) + 1/2*lnx - 1/4*ln(1+x^2) + C
= 1/2*lnx*x^2/(1+x^2) - 1/4*ln(1+x^2) + C