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Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ $$$$ I was given this integral by a friend who saw this here on MSE. He asked me if I could solve it using the very basic tools I have. I was of course unsuccessful.$$$$

My short and unsuccessful attempt: $$\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $$

$$\int_0^1\left(\frac{1}{(\ln x)^2} + \frac{1}{(1-x)^2} +\frac{2}{\ln x(1-x)}\right) \mathrm dx $$ $$\int_0^1\frac{1}{(\ln x)^2}+\int_0^1\frac{1}{(1-x)^2}+\int_0^1\frac{2}{\ln x(1-x)}$$

The problem (apart from the fact that I don't know as yet how to attempt the third integral) is that the second integral (I think) diverges.

Could somebody please help me solve it using preferably only these: Integration by Parts, Integration by Substitution, Partial Fraction Decomposition or Differentiating under the Integral Sign?$$$$ Thanks so much in advance, and I'm truly sorry for the trouble I have caused.

User1234
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    If this would help, the question already exists here. – User1234 May 27 '15 at 14:29
  • I know of no normal, standard high school that includes these things in its curriculum. Perhaps some high school for very gifted students... – Timbuc May 27 '15 at 14:38
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    One thing for sure: splitting into three definite integrals won't work, since each of the three, evaluated by itself, diverges. – Barry Cipra May 27 '15 at 14:41
  • Sir, what would you suggest then? – User1234 May 27 '15 at 14:45
  • @jack is this really a duplicate? going over the old topic there is nothing even close to high-school maths. so if someone would find something which fits this requirement it would be really "new" – tired May 27 '15 at 15:19
  • @tired: I really doubt this problem can be solved through something more elementary than the robjohn's approach in the other question. Even if I am wrong, I think it is best to add another solution to the other thread, just to keep things tidy. – Jack D'Aurizio May 27 '15 at 15:48
  • @jack ok, you're right! – tired May 27 '15 at 15:55

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