# Math Help - Definite Integration

1. ## Definite Integration

Hi,

Could somebody please help me in getting a closed form expression for integrating the following expression:

(1+(x^m))^n

limits of integration are from 0 to 1. Integration is to be done with respect to x and m and n are positive real numbers.

Is it to do with the incomplete beta function, but the upper limit is coming out to be -1, which should lie in the interval 0 to 1.

Thanks in anticipation.

Dekar

2. It appears to be hypergeometric. I ran it through Maple and it gave me.

$\int{(1+x^{m})^{n}}dx=xhypergeom\left([\frac{1}{m},-n],[1+\frac{1}{m}],-x^{m}\right)$.

3. Thanks Galactus,

Can you please help me in getting the expansion of this function?

Thanks

Dekar

4. Originally Posted by dekar
Thanks Galactus,

Can you please help me in getting the expansion of this function?

Thanks

Dekar
See here.

-Dan

5. Originally Posted by dekar
Hi,

Could somebody please help me in getting a closed form expression for integrating the following expression:

(1+(x^m))^n

limits of integration are from 0 to 1. Integration is to be done with respect to x and m and n are positive real numbers.
I think the following works for $n,m>0$.

$\int_0^1 (1+x^m)^n dx$
Let $t=x^m \implies t' = mx^{m-1}$
Thus,
$\frac{1}{m}\int_0^1 (1+t)^n t^{-1 +\frac{1}{m}}dt= \frac{1}{m}\mbox{B}\left( n+1, \frac{1}{m} \right)$

6. Hi,

It won't work because it is (1+t), to have it to get reduced to the beta function we need (1-t),

Thats why I guess, It may go to Incomplete Beta Function if we substitute say p=-t at this stage, but the upper limit then becomes -1.

Dekar

7. Originally Posted by dekar
Hi,

It won't work because it is (1+t), to have it to get reduced to the beta function we need (1-t),

Thats why I guess, It may go to Incomplete Beta Function if we substitute say p=-t at this stage, but the upper limit then becomes -1.

Dekar
Yes, I made a mistake.
You can see more here.

8. So isn't there a better (a less messy) way to arrive at the closed form solution other than that hypergeometric?

I need it a program friendly way to code this.

Any help is highly appriciated.

Thanks

Dekar