Alilouche A.

# Very complicated integral

I'm looking for a solution to the following integral:

int (1/sqrt[1 - x^2 + a (x^6 - x^12)]) dx

where a is a parameter.

Could you help me to find a solution?

## 4 Answers By Expert Tutors

By:

Andre W.

As an addition to Kirill's method: the substitution x=sin(t) only works for |x|≤1. At x=±1, the integral becomes improper and needs to be split. For |x|≥1, you could use another trig substitution, perhaps x=sec(t).
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10/31/13

Alilouche A.

In fact, I forgot a parameter in the formula, so the integral becomes:
int (1/sqrt[1 - x^2 + a (2(bx)^6 - (bx)^12)]) dx

I'm sorry for this error.
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11/01/13

Andre W.

So now your potential is U(x) = a (2(bx)6-(bx)12). You should also mention that a and b are very small positive numbers. The equilibrium point now is at x0=1/b. Since x=1/r, b is the (stable) equilibrium distance between the interacting atoms/molecules, usually a few Angstroms. With U(1/b)=a, U'(1/b)=0, and U''(1/b)=-72ab, you get the quadratic approximation U(x)≈a(1-36 b(x-1/b)²) and can find the small frequency oscillations about the equilibrium.
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11/01/13

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Friendly tutor for ALL math and physics courses

Alilouche A.

I think an analytical approach is more significant from the physical point of view, especially highlighting the phenomenons of orbiting and spiraling of the particles
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11/01/13

Andre W.

If you find an analytic solution, you should publish it in JMC (Journal of Mathematical Chemistry)!
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11/01/13

Tutor
4.4 (10)

Experienced scientist, mathematician and instructor - William

Alilouche A.

I look for a solution for this integral which contains a singularity for some points
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11/01/13

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