Fundamental Theorem of Calculus

1. Nov 13, 2008

Pacopag

1. The problem statement, all variables and given/known data
The FTC states that
$${d\over{dx}}\int_a^x f(t)dt = f(x)$$
Now, how do I do something like
$${d\over{dx}}\int_a^{g(x)} f(t)dt = ?$$

2. Relevant equations

3. The attempt at a solution
I know that it has to do with the chain rule, but I forgot my textbook at school and I can't seem to find it online (e.g. Wikipedia).

2. Nov 13, 2008

Staff: Mentor

Changing the variable to u, where u = g(x), the integral looks like this:
$${d\over{du}}\int_a^u f(t)dt = f(u)$$

The trouble is, you want $${d\over{dx}}\int_a^u f(t)dt = f(u)$$

So if you want d/dx(H(u)), that's the same as d/du(H(u))*du/dx, isn't it? (Here, H(u) represents the value of the definite integral."

3. Nov 14, 2008

Thanks!