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Context

This is a problem in my friend's textbook. The course covers basic calculus, with an introduction to limits, derivatives and integrals. This is important, because when I say "solve this integral without using substitution", I mean before learning substitution. Otherwise, one might be tempted to use more advanced tools (like integrating the Taylor expansion instead), rather than less.

Problem

Evaluate $\displaystyle\int\frac{2x}{x^2+1}\mathrm dx$

Caveat

Being that this is before the chapter teaching the substitution method, they seem to be hinting at some method of solving this even more basic than that.

My thoughts so far

There was a problem in an earlier chapter, namely to evaluate $\displaystyle\frac{\mathrm d}{\mathrm dx}\ln(x^2+1)$, and lo!

So the only method I can think of, that is even basic-er than substitution, is to remember that you have differentiated this function before and recall that the integrand above is the result of that derivative.

...and then add $C$ (lest I forget, again).

Question

So, assuming the idea was NOT to simply recall and remember as if one has an eidetic memory, do you have any ideas on how this can be solved in a more basic fashion than using substitution? I'm at a loss.

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    $\begingroup$ I don't think there are any techniques that are more basic than substitution but less basic than inspection. $\endgroup$ – preferred_anon Jan 11 '18 at 9:48
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    $\begingroup$ Before I learnt substitution, I just remembered $\displaystyle\frac{\mathrm d}{\mathrm dx} \ln (x) = \frac{f'(x)}{f(x)}$. I don't recall there being another way to integrate without substitution. $\endgroup$ – Landuros Jan 11 '18 at 9:50
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    $\begingroup$ May be you stress out what you have learned so far. So that one can give an appropriate response to your inquiry $\endgroup$ – Guy Fsone Jan 11 '18 at 9:52
  • $\begingroup$ It would be what your thoughts were i.e. being aware of the derivative of $\ln (x^2 +1)$ $\endgroup$ – frog1944 Jan 11 '18 at 9:54
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    $\begingroup$ Ultimately, all integrals will eventually come to a form where you just "remember" that you've differentiated something before and gotten the integrand as a result. You can't really solve integrals with pen and paper without it. Exactly where you take that step is, of course, different from problem to problem, but I wouldn't be opposed to that technique on a general basis. $\endgroup$ – Arthur Jan 11 '18 at 9:54
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Considering that your attention is drawn to the fact that the denominator is $D(x) = x^2 +1$, and that you have to solve $$ I = \int\frac{D'(x)}{D(x)}\mathrm dx $$ Now you may have learnt by heart (or you may easily deduce) that for all integrals of this kind, $I = \ln D(x) + C$. Hence you find that the solution is the logarithm of the denominator, plus an integration constant.

Now there's a matter of interpretation:

  • have you substituted the denominator by a function,

  • or have you exploited that you know that all integrals of the given kind have the solution $I = \ln D(x) + C$, and you realized that your integral IS of that given kind?

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