Derivative of an integral

**rincewind2** · February 5th 2010, 06:07 AM

Hi everyone,
First of all, great forum! Hope someone out there can help me : )

Here`s the problem:

Let $f(x) = sin( \frac{1}{x}) for x \neq 0, f(0) = 1$

Calculate the derivative of $F(x) = \int_0^x f(t)dt$

**felper** · February 5th 2010, 06:17 AM

You have to use the Calculus Fundamental Theorem: $\left( \int_0^x f(t)dt \right) '=f(x)$

**rincewind2** · February 5th 2010, 06:31 AM

You mean the answer is just: $sin( \frac{1}{x} )$ ?
(That`s way too easy)

Anyway, thanks a lot for the quick reply : )

Ok, one more question, so I don`t open a new thread:
How can I proof that f(x) is integrable over every interval [a,b]?

EDIT:
I guess this helps a lot xD
"If f(x) is bounded on [a,b] and has finitely many points of discontinuity on [a,b],
then f is integrable on [a,b]."

**buckeye1973** · February 5th 2010, 06:40 PM

Originally Posted by rincewind2

You mean the answer is just: $sin( \frac{1}{x} )$ ?

I think there's more to it - note that you are doing a definite integral from 0 to x. Do you know how to plug in your limits of integration into your integral? Remember that they told you what f(0) is, since sin(1/0) is undefined.

Brian

**rincewind2** · February 5th 2010, 11:47 PM

Originally Posted by buckeye1973

Do you know how to plug in your limits of integration into your integral?
Brian

I`m sorry, but no : (
Can you explain more... I really need this one

**buckeye1973** · February 6th 2010, 07:59 PM

Originally Posted by rincewind2

I`m sorry, but no : (
Can you explain more... I really need this one

You are going to need to know how to calculate a definite integral to solve this problem, and if you are doing calculus, you really ought to know how to calculate a definite integral! This problem doesn't make sense without it.

EDIT: Sorry, as I wasn't thinking right when I gave this answer, Vince's answer below is better.

**vince** · February 6th 2010, 08:06 PM

$\frac{d}{dt} \left( ~\int_{a(t)}^{b(t)} f(x,t) ~dx ~\right) = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t}(x,t) ~dx + f(b(t),t) b'(t) - f(a(t),t) a'(t),$
so in your case the answer is what felper said. I'm very sure.

On-Edit: Everything on the right hand side in the equaiton above is 0 except the second term which is your answer. there is no need to worry about the limits because you're differentiating with respect to x not t. Had the integrand been a function of both x and t, then you would have had to worry about the limits.

**rincewind2** · February 7th 2010, 12:30 AM

Lol... finally got it xD
Everyone, thank you for the replies, you`ve been most helpfull : )

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