What are some examples of difficult integrals that are done using substitutions?

For example: $$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$$

Please no laplace and fourier transforms as I haven't studied those yet.

  • $\begingroup$ Just to clarify: are you looking for examples of these sort of integrals, or are you looking for help with this integral? $\endgroup$ – apnorton May 9 '13 at 15:04
  • 2
    $\begingroup$ I know how to do this, I am just looking for more integrals. $\endgroup$ – please delete me May 9 '13 at 15:09
  • $\begingroup$ How do you do this one? Wolfram can't get an answer in terms of elementary functions $\endgroup$ – imranfat May 9 '13 at 18:37
  • $\begingroup$ divide both numerator and denomerator by x^2 then use the substitution u=x+(1/x) $\endgroup$ – please delete me May 10 '13 at 0:34
  • $\begingroup$ I'd like to see the details of how your example is solved. $\endgroup$ – John Adamski Mar 11 '15 at 19:49

Here is a thread from AoPs that contains a good collection of indefinite integrals. You can find several non-trivial integrals done using substitution.

Indefinite Integral Marathon


I can't tell if it is at the accepted answer's list, but $$\int\sqrt{\tan{x}}\,dx$$ is a good one. It's pretty concise, and perhaps at first it feels like either it is going to be very easy or not doable with elementary functions. It is doable with elementary functions and techniques, but it takes a fair amount of effort.


Some More Questions $$\displaystyle \int \sqrt{\tan x}dx\;,\int\sqrt{\cot x}dx\;,\int \left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx\;,\int \left(\sqrt{\tan x}-\sqrt{\cot x}\right)dx$$

$$\displaystyle \int\frac{1+x^4}{1+x^6}\;\;, \int\frac{1}{1+x^6}dx$$


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