- #1

- 986

- 9

**Show that .... if equivalent to ....**

## Homework Statement

Show that

∫e

^{-t2}sin(2xt)dt (bounds: 0,∞)

= e

^{-x2}∫e

^{u2}du (bounds: x, 0)

## Homework Equations

**THEOREM X.**Let the integral F(x) = ∫f(t,x)dt (bounds: c, ∞) be convergent when a ≤ x ≤ b. Let the partial derivative ∂f/∂x be continuous in the two variables t, x when c ≤ t and a ≤ x ≤ b, and let the integral ∫∂f/∂x dt be uniformly convergent on [a, b]. Then F(x) has a derivative given by F'(x) = ∫ ∂f(t, x)/∂x dt.

## The Attempt at a Solution

I think I need to differentiate both sides enough times until I've "shown" that the equality is true. So far, no success.

I'd let G(x) = ∫e

^{u2}du (bounds: 0, x). Then, using the fundamental theorem of calculus,

(e

^{-x2}G(x))' = e

^{-x2}* e

^{x2}+ e

^{u2}du * e

^{-x2}.

Some pattern will emerge when I keep on differentiating.

As for ∫e

^{-t2}sin(2xt)dt (bounds: 0,∞), I'll have 2x accumulating over and over again, and then an alternation between sin(2xt) and cos(2xt), etc.....

Am I doing this right? I'm not finding any major cancellations.