Question

Why does the derivative with respect to x on an integral in terms of t but evaluated from 0 to x come out to be the original function but in terms of the derivative(x)?

Answer 1

\[\frac{d}{dx}\int\limits_{0}^{x}e^{t^{2}}dt = e^{x^{2}}\] Thanks!I'll review the fundamental theroem and see if that helps.

Answer 2

I understand this: \[\frac{d}{dx}\int\limits_{0}^{x}e^{x^2}dx = e^{x^2}\] but I don't understand why the variable inside the function has no effect on the outcome.

Answer 3

Well, when you integrate e^t^2 you will get some function of t, we'll call it f(t).

So your intermediate step looks kind of like this: \[\large \frac{d}{dx}[f(t) |_0^x]=\frac{d}{dx}[f(x)-f(0)]=\frac{d}{dx}[f(x)]-\frac{d}{dx}[f(0)]\]

of course f(0) is a constant, and f(x) is really the integral of the function, so by taking the derivative you are getting the same function back again.

However you can now see what might happen if instead they made the problem look like this? \[\large \frac{d}{dx}\int\limits_x^{x^2}e^{t^2}dt\]

By following the chain rule and the steps I outlined above with a little thought you will get the answer.

Answer 4

This makes total sense and is what I had already done. Unfortunaley the answer is just e^x^2 without some constant which is the most confusing part to me.

Answer 5

Here is the original problem: \[\frac{dy}{dx} + 2xy=1 ; \ \ \ \ y = e^{-x^2}{\int\limits_{0}^{x}e^{t^2}dt +c_{1}e^{-x^2}}\]Verify that the function is a solution of the given difeq. If the derivative of the integral evaluated to just e^x^2 it's cake which is where i got stuck.

Answer 6

Really I'm just confused as to how we can be sure that the function evaluated at 0 equals 0 and not some constant.

Answer 7

See, the derivative of a constant is zero.

Answer 8

I am saying f(0) is a constant. But it doesn't matter what it is since \[\large \frac{d}{dx}[f(0)]=0\]

Answer 9

Dammit. I've got too much blood in my caffeine system! Thanks Kainui.

Answer 10

My teacher always said don't drink and derive, I just didn't know he was talking about coffee.