Discovering a General Formula for the Function: Hints and Tips

[greisen]

Hi,

I have to find a general formula for the function

1/(2n)!$$\int_{-\inf}^{\inf}x^{2n}*e^{-ax^2}$$I am a little bit lost in how to proceed - any hints appreciated thanks in advance

 

[tiny-tim]

1/(2n)!$$\int_{-\infty}^{\infty}x^{2n}*e^{-ax^2}$$​

Hi greisen!

(btw, it's "\infty")

I assume you've tried integrating by parts, and found that if you start with the x^2n, it just gets worse, and you can't start with the e^{-ax^2}, because it doesn't have a nice anti-derivative?

Hint: can you split the product in some other way, so that you can get an anti-derivative involving the e^{-ax^2}?

 

[greisen]

Thanks

yes I have tried integration by parts but that got very messy.

I don't quite understand how to rewrite the equation involving the anti-derivative?

 

[sutupidmath]

well, i would say sth like this would put u on the right track

$$\int_{-\infty}^{\infty}x^{2n}*e^{-ax^2} = \int_{-\infty}^{\infty}x^{2n-1}xe^{-ax^2}dx$$ then performing integration by parts by letting

$$u=x^{2n-1},v=\int xe^{-ax^2}dx$$

I think this would work. By performing integ by parts a couple of times, i think you will be able to notice some pattern, then use induction to prove the general case.

P.S. This is basically what tiny-tim's Hint says...i hope at least...lol...

 

[tiny-tim]

problem … solution … it's just a state of mind!

Yutup, sutupidmath is right, of course.

The trick is to recognise the difference between a problem and a solution.

In this case, the problem is that you look at ∫e^-ax^2
and think
"I can't integrate that … if only it had an extra 2ax in front of it!"
and change that round to
"I can integrate that if i put an extra 2ax in front of it!"

Problem … solution … it's just a state of mind!