Evaluate $\displaystyle \int_0^\infty {e^{ - \left( {x^2 + x^{ - 2} } \right)} \,dx}.$
Cheers
Another easy one? Well, I don't have a lot of time right now and I don't want to be a complete killjoy to others, so I'll outline a solution:
Note that $\displaystyle \displaystyle \int_0^\infty {e^{ - \left( {x^2 + x^{ - 2} } \right)} \,dx} = e^2 \int_0^\infty {e^{ - \left( {x - \frac{1}{x} } \right)^2} \,dx} = e^2 \, I(1)$, where $\displaystyle \displaystyle I(\alpha) = e^2 \int_0^\infty {e^{ - \left( {x - \frac{\alpha}{x} } \right)^2} \,dx}$ (this is of course the old trick).
Note that $\displaystyle \frac{dI(\alpha)}{d \alpha} = 0$ and therefore $\displaystyle I(\alpha) = c$, where c is a constant (this is an interesting result). To get c, use the well known result for $\displaystyle I(0)$.
Therefore: $\displaystyle I(1) = ....$ and so .....
Yet another easy one?
No. (And let's not get into infinite recursion here).
Nor is anyone. Which would make for a pretty boring board.
You're probably right. Mind you, my outline still leaves plenty to do ..... And personally, I always think differentiating under the integral is pretty cool. And a neglected technique in most undergraduate calculus courses ......
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I just knew you was gonna say that ..... Well, someone elses turn.
Out of interest, why are you posting these integrals? Are they red rags being waved at bulls? My speculation is that at this time of year business is slow and questions like these liven things up a bit (and have good teaching points as well).
Ditto!
Krizalid, you have a good reputation here. And sometimes you solve the problem yourself immediately after the question. That confuses me
Do you want someone to solve them OR do you want it to be some kind of a tutorial ? If it's the latter you could probably post it under Calculus Tutorials. Ask TPH or someone else to choose your integrals as the Problem of the Week e.t.c. Or at least specify it while posting.
Now if it is the former, you probably want someone to solve it in a different way(Given the fact that you always know the solution). If so, then I suggest you specify the methods you have tried. This will stop causing confusions.No poster(helpers I mean) would want their time to be wasted,right?
Naturally , I am sure, all of us appreciate a good mathematical discussion. So I request you to be clear so that we can have a healthy "integration" debate and not quarreling in such a nice forum.
Hope you understand what I mean
Thank you
Iso
Yes, it's nice. For this problem I wanted a solution without it.
Yes, probably the teacher wants a "clean" solution.
Just for fun. I already know a solution for it, so I just post it right here if someone else wants to give it a try.
No, read above.
Well, if a different approach is posted, that'd be nice.
I'm just posting problems like proposed ones.
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I hope I've clarified doubts about integrals
P.S.: in fact, discussion's title is related to the solution that I'm lookin' for.
Well, by setting $\displaystyle x=\frac1u$ leads an useful integral to solve the problem.
Now
$\displaystyle \int_0^\infty {e^{ - \left( {x^2 + x^{ - 2} } \right)} \,dx} = \int_0^1 {e^{ - \left( {x^2 + x^{ - 2} } \right)} \,dx} + \int_1^\infty {e^{ - \left( {x^2 + x^{ - 2} } \right)} \,dx} .$
The rest follows.