in no way i couldn't tackle this problem. could you help me please. and i wonder is there any application/pc program which is solving limit derivative and integral. Like MathXpert. any help will be appriciated.
Welcome to MHF, mami!
You integral does not have an anti-derivative using only standard functions I'm afraid.
But you can calculate it for instance with the online WolframAlpha calculator:
If you know the answer is something clean as then this can probably be done analytically.
One way to go about this is to expand the integrate in a series, integrate each term and sum it up. (Technically, you should prove that you can interchange the sum and the integral, and there are many conditions/theorems to help you with that. But this is the calculus forum so don't worry about that. If it does not work try another "trick".) It may help to expand and first, and then work out their products. I have not worked it out explicitly, but numerics suggest it should work.
Working with series can get quite tedious, so I'll think about a more clever trick.
Is this a problem from a book?