# Integration formula?

1. Dec 11, 2013

### hash054

Integration formula??

I am a graduate student and during my research I have come across this integration formula shows in attached image file. Can anyone help me make sense of this equation because i couldnt find any help from the literature regarding this equation.
Li and Lj are shape functions in this equations whose values Li= ( xj - x ) / ( xj - xi ) and Lj= ( x - xi ) / ( xj - xi )

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Last edited: Dec 11, 2013
2. Dec 11, 2013

### Staff: Mentor

Here's your integral, slightly modified (using a and b as limits of integration rather than the single l (letter 'l') of your thumbnail.

$$\int_a^b L_i^{\alpha}~L_j^{\beta}dl = \frac{\alpha ! \beta !}{(\alpha + \beta + 1)!}l$$

3. Dec 11, 2013

### hash054

thanks for replying .. I know this is the integral.. I was asking about how can we transform it into this factorial form.. any help in that regard?

4. Dec 11, 2013

### Staff: Mentor

I put that in so people wouldn't have to open your thumbnail in another window.

Can you tell us any more about these shape functions? I'm assuming that i and j are indexes and alpha and beta are exponents. What are xi and xj?

Some context as to where this formula came up might be helpful as well.

5. Dec 11, 2013

### SteamKing

Staff Emeritus
6. Dec 11, 2013

### Staff: Mentor

This is almost certainly going to involve the gamma function. Substitute y = Lj, and (1-y)=Li. I assume that it is being integrated between y=0 and y=1. Get Abramowitz and Stegan, and look up gamma functions. The integrals in terms of y are likely to be in there.

7. Dec 12, 2013

### SteamKing

Staff Emeritus
As a matter of fact, it does. You can find several copies of Abramowitz and Stegun online with Google.

8. Dec 12, 2013

### hash054

thanks guys for replying.. I am grateful.. yet I have not taken a course in which gamma functions were included so a little help in evaluating eq. 6.16,17 would be appreciated!!

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9. Dec 12, 2013

### Staff: Mentor

No problem. You need to get yourself a math book that covers gamma functions. Probably Kreyzig would have it; check out the table of contents on amazon. Otherwise, google gamma functions.

10. Dec 12, 2013

### hash054

Got it... the equation 6.16,17 are derived from beta function β (z,w).. which has a relation with gamma function and ultimately in terms of factorial.. The books you guys recommended worked for me!! thanks for the help.. now i can continue!! :)