Very complicated integral

Question

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?
please!

Answer 1

Try substitute x=sin(t); The integral becomes

 

∫dt/√[1+a*sin⁶(t)*(1+sin²(t)+sin⁴(t))];

 

This integral can be evaluated using the following expansion:

 

(1-x)^-½=∑₀^∞(2k-1)!!/(2k)!!*x^k, summed over k. This series converges for x<1.

 

Expression f(t)=a*sin⁶(t)(1+sin²(t)+sin⁴(t)) needs to be less than 1. This puts restrictions on the a parameter. [f(t)]^k is integrable, however, integrals will be very ugly looking. For example,

 

∫f(t)dt=a*

  213 x      337 Sin[2 x]        41 Sin[4 x]      31 Sin[6 x]
{-------  - ---------------     + -------------  -  -------------   +
  256              12                    256                 1024

7 Sin[8 x]        Sin[10 x]
------------   -  ----------- }
 2048                5120

Answer 2

This integral represents the integral curves φ(r) (in polar coordinates, with x=1/r) for the famous 12-6 Lennard-Jones potential for atomic interactions. It is usually treated quantum-mechanically, not as a classical central potential problem.

You should graph the potential, U= a(x⁶-x¹²), and check it has a stable equilibrium at x₀=2^-1/6, corresponding to circular orbits.Deduce the nature of the other orbits (elliptic, parabolic, hyperbolic) qualitatively from your graph. Then do a Taylor series expansion of U about x₀ up to order x², so you will get a harmonic oscillator potential. Substitute this into your integral and get the well-know harmonic oscillator solutions. I doubt there exists an analytic expression for the unapproximated potential.

Answer 3

If I'm interpreting this correctly, you are looking for the indefinite integral of

 

{sqrt[(1-x² + a(x⁶-x¹²)]}^-1 or [1-x² + a(x⁶-x¹²]^-1/2

 

Am I correct?

Answer 4

The Mathematica Integrator says that the integral of this function probably does not exist.