In my calculus class we were solving: integral{0 to x}(sin(x^4)) via power series. My professor told us we can't solve it by any other means, so I went to try to brute force it anyways using Desmos.
I ended up finding that 2x*cos((x^4)/2)*sin((x^4)/2) fit the graph of the integral seemingly perfectly. Even when testing lists of many random numbers, the values of both functions were equal to 11 decimal places (the max shown).
https://www.desmos.com/calculator/jqviupte6g (I can't post images yet)
Now comes my confusion. If the anti-derivative can't be represented by elementary functions, why is this possible and a perfect seeming fit? In addition to that, When solving the derivative of 2x*cos((x^4)/2)*sin((x^4)/2) by hand, it's not anywhere close to sin(x^4). Is it just because it's from 0 to X? I was under the impression that that was equivalent to an anti-derivative but I'm now questioning that thought.