integral

[nikkor180]

Greetings:

Some time ago I wrote in requesting assistance evaluating integral dx / [cos(x) + constant]^0.5. One member responded with a link to Wolfram's Integrator where I found a solution as shown in the provided attachment. I still have two unresolved issues however. First, I don't know how to evaluate the expression. In particular, can anyone tell me how to interpret F(x/2 | -2/(a-1))?

The second and more critical issue is that I had actually hoped for some guidance as to methodology in evaluating the integral. Apparently evaluation requires knowledge beyond the standard methods of integration. If you would point me toward a resource or website from where I might learn such technique, I will be most grateful.

Thank you.

Rich B.






[attachment=0:1cvnjof7]MSP173719g3i391h77ddhi0000064cedh7034194bgd.gif[/attachment:1cvnjof7]

 

[galactus]

The F represents Ellitpic Integral of the First Kind. Try Googling that.

This is very advanced stuff. To work through it by hand would be a proverbial booger. This is why tech is used.

At one time, tables were used. But it is difficult to find tables for Elliptic Integrals these days.

i.e. \(\displaystyle F(k,\phi)=\int_{0}^{\phi}\frac{1}{\sqrt{1-k^{2}sin^{2}\phi}}d\phi\)

The following one is doable via the Beta function. I can show you this:

\(\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{cos(x)}}dx=\frac{1}{2}B(\frac{1}{2},\frac{1}{4})=2.622\)

But your indefinite integral, with the general constant 'a' and no integration limits proves daunting. I ran it through Maple and the solution was horrendous.

 

[nikkor180]

Thank you kindly.