I have a trigonometric function integral in a known range which the answer is 0/0 - 0/0, what should I do to get a logical answer? Can I use Hopital Rule? How should I approach the problem?
Without knowing the exact integral, it is difficult to know the interpretation of your results. We cannot know if the hypothesis needed by the l'Hospital "rule" are satisfied, particularly that the function must be differentiable on an open interval around the limit toward which you want to compute it!
It is an indefinite integral, and "a logical answer" refers to its definition (what kind of I.I.?). If it converges, the l'Hospital rule may be applied.
Also some strange answer, probably you have to simplify the integrand first.
Recently I had this problem in my research. The conditions for L'Hospitals rule were satisfied. I had to repeat the rule two or three times to finally get a good result that was not indeterminate. You can get varied results so make sure you understand the nature of the functions before you go to the limit.
See all above but first, check your math very carefully, especially your initial mathematical model of what you are trying to analyze. 0/0-0/0 suggests a doubly indeterminate result. One possibility is an inadequate initial mathematical formulation of the situation being modelled.
The cronies are right, you can not close your eyes to using L'Hospital's rule. There are examples where the use of this rule leads to the non-existence of a limit, in fact, there is a limit.
if you could manage to communicate your integral (I'm aware that humans can be in conditions, e.g. illness, that don't allow them to follow iterated requests from strangers) I would feed the expression to the program 'Mathematica' and let either you or the community know the result.
If the conditions of Lompital's lemma are not satisfied, then it can not be used. There are examples in which the limit exists, but the use of this rule leads to no existence of a limit.
To use L'Hopital's rule, you first combine the difference between ratios to obtain a single ratio (like combining fractions) of numerator divided by denominator. Depending on the example, this step alone might remove difficulties. But if the resulting ratio still has the property that numerator and denominator both approach zero, then use L'Hopital's rule.
Mathematica does not agree with Issam, as from Issam's third formula-line a factor 'b' dissapeared. As the appended pdf-file shows the result for each of the Integrals is simply 0.
I hope that all contributors who tried to give advice without knowing what the issue really is, will recognize the vanity of such an approach.
sorry for having edited my answer during the creation of your's. So, as you can see, I meanwhile invested the little effort you asks for. You simply left out a factor b and thus got a wrong result. Of course, you are right by saying that Elementary calculus can do the job.
However, as the present example shows, it is not always easy to avoid lapses. Mathematica is not creative enough to make lapses. I found it also extremely useful in symbolic computations (such as in our example) - the older I get, the more so!
Thanks for being given the opportunity to answer your question, giving the condition of the task.
As I guessed in the answer one week ago, there is no uncertainty in your example (and no need in l'Hospital rule). Assuming that the integration limits are from 0 to b, all the terms should be divided into two groups:
n \neq q and n = q. The integrals of the first one members (as indicated by Ulrich Mutze) take zero values.
But for the second group, the average value of the function (sin^2 (z)) is non-zero, i.e. 1/2 times the length of the interval. Therefore, the answer is: b/2 \sum{c_{nn}}. Hope, this helps.
P.S. Mathematica is my friend, but my greatest friend is mathematics (no sarcasm).
you have simply to do the computation right (without omitting factors and re-introducing them in an illegal manner) to see that Amir's expression is simply a sum of zeros and thus 0.
now your calculation is correct. Actually it was so from the very beginning, but was oscured by an unhappy typo. Your comments on this made me think that you follow a queer logic. Actually I misunderstood you. Nevertheless, stating the final result for Amir's expression would have been adequate.
For me the discussion was beneficial since it reveiled a malfunction of Mathematica that will communicate to Wolfram Research.
I got the impression that the question was a request for help. People that ask for help should be given help. Not insults. Although my attempt was not helpful, I am very happy that some other people provided the needed help. So RG does work, even if there are a few members that are less constructive.