[Imustaskforhelp]

Dude I am not joking but today was the day that we were introduced to indefinite integration as a formal chapter in maths at my coaching and we did secx integration.

Basically our sir told us to multiply / divide by sec + tan and observe that its becoming something like integration f(x)^(-1) f'(x) * dx and if we let f(x) as t and this f'(x) * dx becomes dt Actually we can also prove the latter and I had to look at my notes because I haven't revised them yet but its basically f(x) = t

so f'(x) = dt/dx so f'(x)* dx = dt then we get

so integration f(x)^n * f'(x) * dx = integral t^n * dt (where t = f(x)) integral t^-1 dt so we get ln(t) and this t or f(x) was actually sec x + tan x so its ln(sec + tan) and in fact by doing some cool trigonometry we can say this as ln(tan(pi/4 + x/2)) + c

also cosec x integration is ln(tan(x/2)) + c

I haven't read the article but damn, HN, this feels way too specific for me LOL.

[Imustaskforhelp]

So I just started reading the article and it seems that it mentions a point about teachers telling their students to verify it by differentiating the value of integral of secx ie. ln(| tan x + secx|) and it equals secx

and in fact our sir himself told us that he would've also let us do this if we were in normal batches (we are in a slightly higher batch, but most students are still normal and it was easy to digest to be honest except when I was writing this previous comment, I actually found that our sir had complicated the step of f'(x) = df(x)/dx by letting us assume f(x) as t and so on..,maybe it makes it easier to understand considering f(x) to be its own variable like t instead, but that actually confused me a little bit when I was writing the previous comment) , still nothing too hard.

I actually want to ask here because I was too afraid to ask this to sir, but is there a way, a surefire way to solve any integral , like can computers solve any integral?

[purplehat_]

Remarkably, there isn’t a way to solve most integrals symbolically. We say that the set of “elementary functions”, i.e. ordinary looking symbolic functions, is not closed under integration. Even if you try to add special functions in you cannot feasibly make it closed under integration. I’ll try to write something more detailed later but in the meantime you should look up Liouville’s theorem and non-elementary antiderivatives.

[thelaxiankey2]

symbolically no (in fact I believe it can be proven that it's impossible)

numerically sure (ie definite integrals can be evaluated for given values)

[ForOldHack]

Which is course leads to the misnomer in the title: The integral was long solved by numeric means, more easily so with the inventions, but the proof of the solution, took a while... and as some other brilliant hacker-newsestition, pointed out, it because even easier with an ingenious u-substitution, related to the solution of the integral of 1/x discovered in the late 1930s...

( The solution is both possible, and proved, and there is a goddamned youtube video about the trick, and its not a minor trick either, like the proofs of int (sec (x)) or int (1/x ). )

In my text book, and current text books, it is said it cannot be resolved by elementary means, and it cannot, but it can be solved, and proven by one whopper of an idea.

The research is left as an exercise.