# Thread: Derivative of an integral

1. ## Derivative of an integral

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

Heres the problem:

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

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

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

3. You mean the answer is just: $\displaystyle sin( \frac{1}{x} )$?
(Thats way too easy)

Anyway, thanks a lot for the quick reply : )

Ok, one more question, so I dont 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]."

4. Originally Posted by rincewind2
You mean the answer is just: $\displaystyle sin( \frac{1}{x} )$?
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

5. Originally Posted by buckeye1973
Do you know how to plug in your limits of integration into your integral?
Brian
Im sorry, but no : (
Can you explain more... I really need this one

6. Originally Posted by rincewind2
Im 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.

7. $\frac{d}{dt} \left( ~\int_{a(t)}^{b(t)} f(x,t) ~dx ~\right) = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t}(x,t) ~dx + f(b(t),t) b'(t) - f(a(t),t) a'(t),$
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.

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