They do exist.

I used to think I was an integration hot-shot until I came across e^(e^x). Go on, have a go. You can throw everything you've got at it - substitution, integration by parts... - but you just make it worse, like picking at a scab.

Please note my correct use of the ellipsis.

You CAN, however, approximate a solution to this bastard by using a Taylor expansion. Another impossible integral: 1/(ln(x))

A possible but pretty tough one, good practice: sqrt(tan(x)). And remember, you don't learn anything by asking Maple.

later: Correction, you learn that Maple makes a big ugly mess attempting to integrate sqrt(tan(x)). Ask it to simplify(%); its answer for a laugh.

I never thought I was an integration hotshot (probably partly because my high school class valedictorian is currently one of the top four people in the state in calculus in Mu Alpha Theta competitions), but I know of a couple more that I'd like to add.

First, e^(-(x^2)). Try whatever you want with it as an indefinite integral; nothing works. It is, however, approximable with Taylor and MacLaurin expansions, as well as more mundane methods.

Another that is at least significantly difficult and possibly impossible is sqrt(x)sec(x). I've only tested this on my TI-89, which is much more simplistic than any computer mathematics software one would try these on. If anyone finds this to be either possible or impossible, /msg me, and I'll update this.

When I started writing this w/u, I thought I might have found a heuristic for at least some impossible integrals, but sqrt(tan(x)) disproves it. My false heuristic is that f(x) has one or more jump discontinuities and a bound on the domain. Impossible integrals that meet this heuristic still may be far more common than those that do not, although this prediction is to the zeroth approximation.

There are certain integrals that cannot be expressed in terms of elementary functions, which is why special functions have been defined for some of them. The integral of exp(-x2) is (give or take a few factors) defined as the error function. The integral of 1/ln(x) is the logarithmic integral function. Other "impossible" integrals include that of sin(x2) and cos(x2) which are the Fresnel integrals, and integrals that involve the square root of a cubic or quartic polynomial (elliptic integrals).

However, the fact that there exist unmeasurable sets for any measure theory means that there also exist functions that cannot be integrated under any definition of the integral, be that Riemann's definition or even the more advanced Lebesgue integral. Clearly, one cannot hope to integrate the characteristic function of such an unmeasurable set, as the value of its integral would correspond to the measure of that precisely unmeasurable set.

Many definite integrals are made much easier by taking the domain to be complex, making up an appropriate contour, and performing contour integration with the aid of complex analysis (Cauchy's integral theorem, residue calculus, Jordan's lemma, etc.)

Not suprising, as complex analysis was invented in part to tackle all the gross integrals coming out of fluid mechanics, electromagnetism, and other scientific fields in the 1800s...

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