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Q&A

Why can an easily graphable definite integral, be labyrinthine to evaluate?

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How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graph looks so unremarkable and straightforward?

I used Desmo to graph $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} , dx$ below. u/camelCaseCondition's comment reaffirms that the solution is knotty and effortful, but not why:

The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least be understandable had you taken Calc I-III and Complex Analysis.

However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]

My question is more specific than Why do statements which appear elementary have complicated proofs? and Why are mathematical proofs so hard?, because I'm asking especially about definite integrals with uneventful graphs, but that high schoolers can effortlessly graph without software.

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