# Thread: Equivalence of two integrals

1. ## Equivalence of two integrals

Ok, so the question I'm looking at involves the following integral;

$\displaystyle I[y] = \int_{-\infty}^{\infty} (y'^2 + x^2y^2)\, dx$

restricted to functions with

$\displaystyle \int y^2\, dx = 1$

I'm trying to show that under the (given) assumption xy^2 --> 0 as x --> +infinity
we can write

$\displaystyle I[y] = 1 + \int_{-\infty}^{\infty} (y' + xy)^2\, dx$

I've tried integration by parts, taking out the y'^2 and various other things but get the feeling I'm missing something really obvious that allows me to do the question.
Help?

2. ## Re: Equivalence of two integrals

Originally Posted by eigensheep
Ok, so the question I'm looking at involves the following integral;

$\displaystyle I[y] = \int_{-\infty}^{\infty} (y'^2 + x^2y^2)\, dx$

restricted to functions with

$\displaystyle \int y^2\, dx = 1$

I'm trying to show that under the (given) assumption xy^2 --> 0 as x --> +infinity
we can write

$\displaystyle I[y] = 1 + \int_{-\infty}^{\infty} (y' + xy)^2\, dx$

I've tried integration by parts, taking out the y'^2 and various other things but get the feeling I'm missing something really obvious that allows me to do the question.
Help?
What were the details of the integration by parts that you used?

3. ## Re: Equivalence of two integrals

Ok, so I started by taking the y'^2 out into a separate integral since it also appears in what I want to get.
Then I'm left with an integral of x^2y^2 but the difficulty I've found arises after I've chosen which part to take as u and which as dv since eg. if dv = y^2 then I can't find v. I think I must have tried just about every combination of x, x^2, y, and y^2 but always either end up with a more complicated expression or one it's impossible to find.

Are you saying that integration by parts is what I should be using?