My doubt is how to behave when we do complex substitutions in integrals. I'm interested now in one example in particular that is $$\int_0^\frac{\pi}2 x^3\csc x\,\mathrm{d}x$$ and my way to solve it is pretty the same as this but trying to solve manually the complex stuff is quite prohibitive. In an another discussion, linked also in the previous one, it is said that the imaginary parts are to be cancelled out because they sum up to zero without giving further explanations. Is there a reason why we left imaginary stuff apart in these cases? The question arises also because I remember other integrals where complex terms are not simplified to their real parts at all… So, definitely, how to know when we can just ignore imaginary parts in the evaluation of real-function integrals?
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