# Integration - looking for closed form

• January 22nd 2008, 11:01 PM
graticcio
Integration - looking for closed form
For the following two integrals,

1. $\int_0^1 \frac{1}{1-x^b} dx$

2. $\int_0^1 \frac{x^{ab}}{1-x^b} dx$

is there any closed forms in terms of (Star)trigonometric functions(Star) ???

thanks for helping
• January 22nd 2008, 11:16 PM
mr fantastic
Quote:

Originally Posted by graticcio
For the following two integrals,

1. $\int_0^1 \frac{1}{1-x^b} dx$

2. $\int_0^1 \frac{x^{ab}}{1-x^b} dx$

is there any closed forms in terms of (Star)trigonometric functions(Star) ???

thanks for helping

If you give values to a and b and run the result through the Wolfram integrator, I think you'll see the difficulty if not impossibility of a closed form solution in terms of trig functions.

Eg.
• January 22nd 2008, 11:24 PM
graticcio
yes, but there appears to be a pattern...
some arctan and some log...

can we use any trigonometric substitution to evaluate?

thanks
• January 23rd 2008, 12:00 AM
mr fantastic
Quote:

Originally Posted by graticcio
yes, but there appears to be a pattern...
some arctan and some log...
[snip]
thanks

Yes, that's probably due to the underlying hypergeometric function .....
• January 23rd 2008, 01:20 AM
graticcio
so what is the relationship between hypergeometric function and trigonometric function in this case ???

or how to represent the hypergeometric function here by trigonometric functions?

thanks
• January 23rd 2008, 01:27 AM
wingless
Quote:

Originally Posted by graticcio
1. $\int_0^1 \frac{1}{1-x^b} dx$

2. $\int_0^1 \frac{x^{ab}}{1-x^b} dx$

I think those functions don't converge on the bounds 0 to 1. So the integral doesn't exist.
• January 23rd 2008, 01:48 AM
graticcio
they converge with x<1
• January 23rd 2008, 02:08 AM
wingless
Quote:

Originally Posted by graticcio
they converge with x<1

On second thought, you're right..