Numerical vs. Symbolic

Wolfgang Tintemann asked me how to compute

\(\int\limits_{1-\sqrt2}^{\sqrt2-1} \dfrac{\sqrt{|1-6x^2+x^4|}}{(1+x^2)^2} \,dx\)

in Euler Math Toolbox. The aim was an „exact solution“, of course. Now I am not in charge for the Maxima subsystem contained in Euler, which in fact cannot „solve“ this. But the point is that I strongly doubt, that symbolic solutions are useful at all.

Let us try Wolfram Alpha.

fint

Okay, this is at least some output. It involves the inverse sine function and the elliptic integral of second and first kinds. Not much is gained by this.

The function itself does not look too complicated.

f

The definite integral in question is the inner part, which is easily done with numerical methods like the integrate() function of EMT which uses an adaptive Gauß integral. The result is 0.599070117368…

However, numerical answers sometimes do not provide answers to obvious questions. Often they do, but only with much effort. One question here might be, if the integral over all x exists or not. To answer this, you need to get an estimate of the behavior of the function towards infinity. This is an exercise. The result is O(1/x^2). So the integral exists.

Again, I tried Wolfram Alpha for a solution.

t

That is not helpful.

Since the function is not analytic in the upper half plane, we cannot use residual calculation easily. There might be a way around this, but I did not try too hard to find it.

So we need an estimate. We can prove that

\(f(x) \, x^2 \le 1\)

for x large enough. So we can integrate to a large value and use this estimate to estimate the complete integral. Or we can substitute x=1/t and integrate towards 0. In any case, we get a good numerical answer, I got  2.39628046947…

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