Integration and Leibniz's formula
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For problem #1, its the integral from o to infinity (the symbol for infinity for that problem was cut off)© BrainMass Inc. brainmass.com March 6, 2023, 1:12 pm ad1c9bdddf
See attached file 'Probability_solutions.doc'.
Integrals like this are a bit tricky, so let's look at them in general and see how we can figure them out...
(Incidentally, I'm in Physics, and I use this integral all the time. I had to commit it to memory long ago!)
This is just a Taylor series of e-ax around x=0. Look it up if it's unfamiliar to you.
So - what use is this? Well, it lets us examine the integral in a (slightly) easier way:
Check this last bit. I've just used the expression for e-ax that we got above. Also look at the factorials (!) and check they're right. [I'm assuming you know, but in case you don't x! = (x)(x-1)(x-2).....1. For example, 4! = 4 x 3 x 2 x 1 = 24. Factorials get very big very quickly. ...
This shows how to differentiate for x and y and how to use Leibniz's formula. The infinity integrals are given.