Your favorite "tricky" integrals?

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PM 2Ring
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Re: Your favorite "tricky" integrals?

Postby PM 2Ring » Thu Jan 12, 2012 4:56 pm UTC

Afif_D wrote:
dissonant wrote:This one is cute. Find the area bounded by the positive x-axis and the line [math]e^{-x} + e^{-y} = 1[/math]


Are you sure the area is bound and finite?


The graph doesn't look promising, but Wolfram Alpha gets a finite value for
the integral of -log(1-e^(-x)) from x=0 to infinity:
Spoiler:
zeta(2) = pi²/6

Sagekilla
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Re: Your favorite "tricky" integrals?

Postby Sagekilla » Thu Jan 12, 2012 8:28 pm UTC

Afif_D wrote:
dissonant wrote:This one is cute. Find the area bounded by the positive x-axis and the line [math]e^{-x} + e^{-y} = 1[/math]


Are you sure the area is bound and finite?


It is. The area is pi^2 / 6. Solve it as a single variable integration and you'll see.
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Ended
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Re: Your favorite "tricky" integrals?

Postby Ended » Thu Jan 12, 2012 8:36 pm UTC

Yes, quite cute!
Spoiler:
Use the substitution u = e-x, and expand log(1-u) as a Taylor series.

This reminded me of a similar idea (but in reverse) used in proof 1 here [pdf]: http://secamlocal.ex.ac.uk/people/staff ... /zeta2.pdf. The series for zeta(2) is converted into a rather tricky area integral which is evaluated directly.
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
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dissonant
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Re: Your favorite "tricky" integrals?

Postby dissonant » Fri Jan 13, 2012 6:01 am UTC

Glad you enjoyed it. I learnt of this from an answer of Hans Lundmark @ http://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-n-1-infty-frac1n2:

Here, zeta(2) is expressed as the limit of the area of the following curious geometric shape...

Image

tomtom2357
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Re: Your favorite "tricky" integrals?

Postby tomtom2357 » Fri Jan 13, 2012 9:55 am UTC

Hey, I have an idea! The area of e-x+e-y=1 is zeta(2) right? So what about the volume of e-x+e-y+e-z=1?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

harun55
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Re: Your favorite "tricky" integrals?

Postby harun55 » Sat Jul 07, 2012 10:27 am UTC

vilidice wrote:One I like:

[math]\frac{1}{6} \int_0^t\ e^{x}(t-x)^3 dx[/math]

a lot of people get hung up on using integration by parts, when it's actually a whole lot easier.

Edit: supposed to be 1/6 not 1/4 (1/3!)


Yes , you are right it take a little time to understand what's the matter, thanks for posting. I discussed this question with my brother who is in high school and he tried partial and than I told him the matter.
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Timefly
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Re: Your favorite "tricky" integrals?

Postby Timefly » Sun Jul 08, 2012 3:52 pm UTC

[math]\int \sqrt{1+sin{x}}\,\mathrm{d}x[/math]

The easy way of finding the above antiderivative is quite nice.


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