Thread: Integration - looking for closed form

1. Integration - looking for closed form

For the following two integrals,

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

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

is there any closed forms in terms of trigonometric functions ???

thanks for helping

2. Originally Posted by graticcio
For the following two integrals,

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

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

is there any closed forms in terms of trigonometric functions ???

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.

3. yes, but there appears to be a pattern...
some arctan and some log...

can we use any trigonometric substitution to evaluate?

thanks

4. 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 .....

5. 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

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

2. $\displaystyle \int_0^1 \frac{x^{ab}}{1-x^b} dx$
I think those functions don't converge on the bounds 0 to 1. So the integral doesn't exist.

7. they converge with x<1

8. Originally Posted by graticcio
they converge with x<1
On second thought, you're right..