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My title is pretty vague, so let me elaborate on what exactly it is that I am looking for.

I came across this integral

$$ \int \frac{x \sin x }{1 + \cos^{2}x } dx $$

Now, I know that this indefinite integral is a monstrosity, and its answer alone is more than half a page long. I checked it on Wolfram Alpha, but what I couldn’t find anywhere was how does one evaluate such an integral in the first place.

In calculus classes we are taught basic techniques like integration by parts and u-substitution, and in the case of definite integrals maybe things like Feynman’s technique. But none of these techniques seem like they would be able to handle this monster. I thought of integrating by parts but then you end up with the indefinite integral of $ \tan^{-1}({- \cos x) }$ How would one handle this type of an integral ?

So basically what I am looking for is a step by step solution to the integral above, and also for techniques/theorems which are not generally taught in introductory calculus classes, but which serve as an indispensable tool for calculating these tough integrals. If you can provide any resources for further information on these topics then that would be great. I would also appreciate resources in genaral which serve as a great source to take a deep dive in the world of integrals.

I also have a follow up question to this - how do WolframAlpha or mathematica figure out these integrals ?

Thank you in advance.

Srinidhi kabra
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  • The reality is, some indefinite integral don't have a known closed form. It is a miracle when they do. In calculus, we are shown examples with closed form solutions using known techniques, but most indefinite integrals don't have solutions. – Thomas Andrews Apr 09 '23 at 13:22
  • @ThomasAndrews How do we know which integrals havw closed forms and which one’s do not ? Isn’t there like any algorithm/technique/theorem which we can use to determine that ? – Srinidhi kabra Apr 09 '23 at 13:35
  • There are theorems that answer sometimes whether an integral will have closed form or not, but certainly not general theorems that answer it all the time. I would guess (without evidence) that the question of whether there is a closed form for an integral is similar to the question of whether a Diophantine equation has a solution, and thus is undecidable in the general case. The best we can do is individual cases. – Thomas Andrews Apr 09 '23 at 13:46
  • For the general question, this may be of interest: https://math.stackexchange.com/questions/70974/lesser-known-integration-tricks – Hans Lundmark Apr 09 '23 at 13:55
  • @HansLundmark thanks a lot, that actually helps. Do you know of any books / online papers / resources that I can go through ? – Srinidhi kabra Apr 09 '23 at 16:57
  • I don't know... You might enjoy Vălean's book: https://www.amazon.com/Almost-Impossible-Integrals-Problem-Mathematics/dp/303002461X – Hans Lundmark Apr 09 '23 at 17:05

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