We can write,
Thus, if you use :
Now, there is a way to get,
It is just really ugly.
~~~
We can use the formula that if then:
This formula can of course be painfully derived by the substitution .
I'd like to initiate an
The rules are simple
I start, here's the first one:
- The person who solves an integral, proposes the another one, but before proposing it, he must have the approval (if the answer is correct or not) of the person who proposed the previous integral.
- Only propose integrals “of one variable”; this is because all do not know multiple integration (I include myself). The level of the proposed ones must be middle-upper.
- Indefinite, Definite and Improper integrals are allowed.
- Use LaTeX
Greetings
Indeed! I forgot the a factor of in my substitution.
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Here is an indirect way of doing this integral.
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Use the same substitution as above to get correctly this time,
Let
Integration by parts,
Now consider the integral,
Use substitution to get,
Let . This integration by parts gives that:
Thus,
i think it was a great idea, and a good opportunity to see some interesting integrals. of course it also defeats the purpose if we get stuck on one integral too long. maybe we can modify the rules a bit to keep things moving. like put a time limit on when the integral should be solved--if not solved within that limit, the person who suggested the limit in the first place should give the solution and choose another integral. and of course, there should be some reward for coming up with an integral that no one can solve (within reason)
Solution:
for .
We use the formula: .
Since :
Integrate term-by-term*
*)This is legal because of something called uniform convergence.
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Note, this integral was taken from here.
Experience. The two most useful techiques in integration are:
1)Power Series
2)Contour Integration
#2 is usually the best. But I do not understand it yet. One of the things I am learning over the summer is in order to under approach #2. If you look in the link I gave, TD!, computes the first integral using that approach.
I think it is a good idea too. But we should try to follow the rules and if no response let the original poster answer the question. I think it will give a chance to let people see some hard/interesting integrals.i think it was a great idea, and a good opportunity to see some interesting integrals. of course it also defeats the purpose if we get stuck on one integral too long. maybe we can modify the rules a bit to keep things moving. like put a time limit on when the integral should be solved--if not solved within that limit, the person who suggested the limit in the first place should give the solution and choose another integral. and of course, there should be some reward for coming up with an integral that no one can solve (within reason)
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What about this integral.