well, i don't know, this might be related to "integration" somehow but this thread is supposed to be about "techniques of integration"! so let's just keep it that way please.
i don't want this thread to go in a different direction. cheers!
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I'm going to bend the rules a little bit and give you a few integrals instead of one. Pick whichever you like and share your solution with us if you feel like it:
1)
2)
3)
4)
5) (the answer to this one is in terms of Gamma function!)
Have fune!
Start with this simple fact: Thus the integral becomes (after reversing integration order) so it remains to compute and its value is which is the answer of the integral.
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See below for full answer. I did this when I was in college, so no time to do anything else!
correct!
and that gives us: which is the correct answer. i hope you'll also post your solutions!
no.4
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Ok, this problem is nice:
Problem: Suppose are continuous, is decreasing and is increasing. Prove that for all real numbers
Note: Here by we mean The solution I know doesn't use any known integral inequalities. Instead, it uses a nice integration technique and it's fairly short!
Since we know that must be non-decreasing - because f is non-increasing- (we can divide by since it is non-zero in our interval), thus we must have
Expand: the inequality will be preserved if we multiply by:
Thus:
Now integrate in and the inequality follows since in -and so the integrals are positive, and we can divide-