I need help to find the integration of the following?

∫ lnx/x dx =
=∫ lnx dlnx = ½ln²x+C

neither is an ideal fraction: x^2 / (x^2-a^2) = a million + a^2 / (x^2 - a^2) 2nd term could be more advantageous as a partial fraction: a^2 / (x^2 - a^2) = C / (x + a) + D / (x - a) a^2 = Cx - Ca + Dx +Da C + D = 0 ----->> C = - D Da - Ca = a^2 Da - (- Da) = a^2 2Da = a^2 2nd = a D = a/2 C = - a/2 a^2 / (x^2 - a^2) = (a/2) / (x - a) - (a/2) / (x + a) combine [x^2 / (x^2-a^2)] dx = combine [a million + (a/2) / (x - a) - (a/2) / (x + a)] dx = x + (a/2)(ln abs(x - a) - ln abs(x +a)) + C = x + (a/2) ln abs[(x - a) / (x + a)] + C =======================================... combine [x^2 / (a million+x^2)] dx = combine [(x^2 + a million)/(x^2 + a million) - a million / (x^2 + a million)] dx = combine [a million - a million / (x^2 + a million)] dx = x - arctan(x) + C

Do you realize how long integration by parts takes?