- #1

- 27

- 0

## Main Question or Discussion Point

I have got this integration:

[itex]\int_{-1}^{1}\frac{x^n}{(1+w^2-P^2-2wx+p^2x^2)^2}dx[/itex]

And at the same time, I can provide several results when n is small.

For example:

when [itex]n=1[/itex], the result is

[itex]\frac{2w}{(1-P^2)(1-w^2)}[/itex]

when [itex]n=2[/itex], the result is

[itex]-\frac{2(1-P^2-w^2)}{P^2(1-P^2)(1-w^2)}+\frac{\Delta}{P^3}[/itex]

when [itex]n=3[/itex], the result is

[itex]\frac{2w(3w^2-2w^2P^2+P^4+2P^2-3)}{P^4(1-P^2)(1-w^2)}+\frac{3w\Delta}{P^5}[/itex]

where

[itex]\Delta=\sinh^{-1}{\frac{P^2+w}{((1-P^2)(P^2-w^2))^{1/2}}}+\sinh^{-1}{\frac{P^2-w}{((1-P^2)(P^2-w^2))^{1/2}}}[/itex]

I have tried it in Mathematica and Maple, However the software can't figure it out.

Does any one feel familiar with this kind of integration and give me some suggestion?

Look forward to your kind reply!

Sincerely Yours

Jacky Wu

[itex]\int_{-1}^{1}\frac{x^n}{(1+w^2-P^2-2wx+p^2x^2)^2}dx[/itex]

And at the same time, I can provide several results when n is small.

For example:

when [itex]n=1[/itex], the result is

[itex]\frac{2w}{(1-P^2)(1-w^2)}[/itex]

when [itex]n=2[/itex], the result is

[itex]-\frac{2(1-P^2-w^2)}{P^2(1-P^2)(1-w^2)}+\frac{\Delta}{P^3}[/itex]

when [itex]n=3[/itex], the result is

[itex]\frac{2w(3w^2-2w^2P^2+P^4+2P^2-3)}{P^4(1-P^2)(1-w^2)}+\frac{3w\Delta}{P^5}[/itex]

where

[itex]\Delta=\sinh^{-1}{\frac{P^2+w}{((1-P^2)(P^2-w^2))^{1/2}}}+\sinh^{-1}{\frac{P^2-w}{((1-P^2)(P^2-w^2))^{1/2}}}[/itex]

I have tried it in Mathematica and Maple, However the software can't figure it out.

Does any one feel familiar with this kind of integration and give me some suggestion?

Look forward to your kind reply!

Sincerely Yours

Jacky Wu