In one sense, the more incredible thing about doing calculus with the elementary functions is that in specific, nice conditions, they give "nice", compact answers not just /at all/, but on a very regular basis, things like 0, 1, 2, 3, 2e, 3π/8, and so on.
Forget everything you know about math for a moment, in order to run a thought experiment. When confronted with some strange, complex formula like the LHS of OP's equation, one that (for our purposes of discussion) you are told is supposed to represent some number, or maybe infinity, or something like that, one would not immediatey expect just by looking at the formula itself that it is in fact equal to some "simple" number like 1, as is in fact the case here assuming OP's formulae are right.
Moreover, staying speculative and artificially naive a moment (when of course we know the following to be true), it is easy to imagine that there an infinity of strange, complex formulae like the OP's which are really really close to some such number, but are in fact not equal to that number.
Recall that (in one treatment) whenever infinity is written, this does not actually invoke the strange notion of infinity-as-such, but instead is a shorthand for a /limit/ of some kind. One is thus obliged to recall the limit laws, and how they may behave with respect to the operators entailed.
From this, one ought also to verify that one actually understands what the integrand is, and what it (literally) looks like, in terms of a graph, if such is feasible. You had better be able to evaluate f(0), f(1) etc, for example, and identify any weird or pathological points, if you propose to integrate.
Not having worked the problem, I don't claim that the above observations lead directly to a proof, or solution. But they are clearly prerequisites, things that you ought to be able to do if you propose to prove such-and-such, which is a more involved thing, and will necessarily entail some of the above at some point.