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Assuming $f(x)$ is a Real function of a Real variable for each "$\sqrt{f(x)^2}$", then as $f(x)$ tends from no crossings of $f(x) = 0$ to chaotic & dense crossings of $f(x) = 0$, does computation of a definite integral remain simple or does it become intractable as the chaos/interval/density increase?

I am looking to better understand how both indefinite and definite integrals of e.g. $(\text{Real})\sqrt{(\cos(1/x) - 1/2)^2}$ are/can be calculated, but I am having difficulty finding any information relating to this type of calculus/tractability research. There are many integral calculators online that return integrals for Real functions such as $\sqrt{(\cos(1/x) - 1/2)^2}$, though they say "steps not shown: steps require too much time" or whatnot, but I'm hoping someone could suggest source & reference material, if not directly answer the opening/title question. What determines whether or not such a function as in the title is integrable i.e. has an indefinite integral?

Winther
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Valentine
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1 Answers1

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Hint.

I'm here answering to your question What determines whether or not such a function as in the title is integrable i.e. has an indefinite integral?

The examples you gave are functions defined and bounded on a compact interval and continuous on the interval except at one point. It can be proven that such function are Riemann integrable.

Now a warning. You use the wording tends to chaotic, which I think is not well define.

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