There are many times when one uses methods from complex analysis to solve integrals that would otherwise be very difficult to solve. My question is:
Can all integrals that are usually solved this way be solved without complex analysis, with the usual integration techniques? Or are there some integrals that can only be solved using complex analysis?
On
The residue theorem is one of the most powerful means of integration.
At the same time, there are strict conditions for its use, and the integration itself sometimes becomes a very difficult task (task1, task2). The second task allows us to compare the complexity of the application of the theorem on residues with the method of decomposition of a polynomial into elementary ones.
At the same time, there are methods that can compete with the theorem on deductions. First of all, integration by parameter.
To my best knowledge, elliptic integrals can not be solved without methods from complex analysis.
Also integrals involving the residue theorem to solve them seem to be hard to solve with other methods but some of them can also be solved without using the residue theorem.
But your questions isn't well-posed: Just because we don't know yet how to solve an integral without complex analysis doesn't mean that there cannot be any way to do so.
You should therefore probably ask: Are there integrals for which the only known ways to solve it are methods from complex analysis?
Also note that if there is a nice way to solve an integral using complex analysis, people will most likely not search for a way to solve these integrals by real (and probably very complicated) methods.