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I have tried to find the integral of $\tan(x^2)$ with respect to the independent variable $x$ but in vain. Substitution doesnt work,integration by parts doesnt work. I have tried expanding $\tan(x^2)$ using expansion formula for $\tan(x)$ and tried integrating term by term but only upto a certain no of terms. So does the integral $\tan(x^2)$ exist and if it exist is there no simple function representing it?

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Unfortunately not all functions made of combinations of $\sin(\cdot)$, $\tan(\cdot)$, $\sqrt{\cdot}$ ... (elementary functions) can have a simple antiderivative, where the antiderivative is also an elementary function. It's also deceptively tricky to check or prove whether your function actually has an elementary antiderivative but it can be proven with the field of math known as Galois Theory. You can read more about the topic in this article on functions with no elementary antiderivative, which includes $\tan(x^2)$.


Antiderivatives of this type are called Nonelementary Antiderivatives and in fact not even $\sin(x^2)$ nor $\cos(x^2)$ have elementary antiderivatives (the solution is defined as the Fresnel Integral).

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  • $\begingroup$ Thank u sir for for clearing my thoughts. $\endgroup$ – Shiksharthi Sharma Nov 17 '16 at 17:33
  • $\begingroup$ @ShiksharthiSharma No problem. I once wasted loads of time trying to find $\int e^{\sin x}$ before I found out it can't be done :) $\endgroup$ – Jam Nov 17 '16 at 17:35
  • $\begingroup$ Could u refer me some textbooks containing about this $\endgroup$ – Shiksharthi Sharma Nov 17 '16 at 17:38
  • $\begingroup$ @ShiksharthiSharma I'm not too sure about which ones would be good. Maybe try one on elementary Galois Theory or asking for textbook recommendations as a new question on this site? $\endgroup$ – Jam Nov 17 '16 at 17:47
  • $\begingroup$ @ShiksharthiSharma Have a look at this Q&A for a start. math.stackexchange.com/questions/265780/… $\endgroup$ – Jam Nov 17 '16 at 18:28

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