Fundamental Theorem of Calculus problem

[Pacopag]

Homework Statement

The FTC states that
$${d\over{dx}}\int_a^x f(t)dt = f(x)$$
Now, how do I do something like
$${d\over{dx}}\int_a^{g(x)} f(t)dt = ?$$

Homework Equations

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).

 

[Mark44]

Homework Statement

The FTC states that
$${d\over{dx}}\int_a^x f(t)dt = f(x)$$
Now, how do I do something like
$${d\over{dx}}\int_a^{g(x)} f(t)dt = ?$$

Homework Equations

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).

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."

 

[Pacopag]

Thanks!