Tricky Integral from Griffiths QM

Click For Summary

Homework Help Overview

The discussion revolves around the integral ##\int_0^\pi \sin^{2l+1}\theta~d\theta##, which falls under the subject area of calculus, specifically focusing on integral calculus involving trigonometric functions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to approach integrals of this type and seeks guidance or resources. A participant questions whether l is a positive integer, which is confirmed by the original poster. Another participant mentions a "reduction formula" for integrals of powers of sine and cosine, suggesting that it can simplify the problem.

Discussion Status

The discussion is active, with participants exploring the nature of the integral and discussing potential methods to approach it. Some guidance has been provided regarding the reduction formula, although there is no explicit consensus on the best approach yet.

Contextual Notes

There is a lack of specific examples or resources mentioned by the original poster, which may limit the discussion. The focus on definite integrals from 0 to pi is noted as a simplification in the context of the reduction formula.

showzen
Messages
34
Reaction score
0

Homework Statement



##\int_0^\pi \sin^{2l+1}\theta~d\theta##

Homework Equations

The Attempt at a Solution



I have no idea how to proceed with integrals of this type, and I can't find any similar examples online.
If someone could give me some info on these types of integral, or direct me towards a resource online, I would greatly appreciate it.
 
Physics news on Phys.org
Is l a positive integer?
 
Yes, sorry I forgot to include that.
 
There's something called a "reduction formula" for integrals of powers of sine and cosine. You can derive it yourself quite simply from integration by parts, or you can just look it up.

This formula is perfect to find elementary indefinite integrals of odd powers of sine (or cosine). Generalizing it to 2l+1 takes a bit more effort, but it's greatly simplified if you only care about the definite integral from 0 to pi.

Edit:
I just used it to find an answer in terms of l, and it works for all cases I've checked (l=1,2,3,4,5, and just to put the cherry on top, l=21)
 
Last edited:

Similar threads

  • · Replies 19 ·
Replies
19
Views
3K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 14 ·
Replies
14
Views
3K
Replies
3
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 34 ·
2
Replies
34
Views
3K
  • · Replies 3 ·
Replies
3
Views
2K
Replies
14
Views
2K
Replies
9
Views
2K
  • · Replies 17 ·
Replies
17
Views
2K