Definition of derivative integration problem

1. Jan 13, 2004

tandoorichicken

This was an extra credit problem on our last test. We haven't learned how to do it yet but I was just curious as to how it would be done.

$$\int^{x^2}_{5} \sqrt{1 + t^2} \,dt = G(x)$$
Find G'(x).

2. Jan 13, 2004

Hurkyl

Staff Emeritus
Try going back to the definition of derivative.

3. Jan 14, 2004

himanshu121

The formula is
$$\int_{f(x)}^{g(x)} \phi (x)dx=\phi [g(x)]g'(x) - \phi [f(x)]f'(x)$$

4. Jan 14, 2004

HallsofIvy

Staff Emeritus
himanshu121, that's an unfortunate notation. It's difficult to distinguish where x is the "dummy" variable and where it is the final variable.

Better would be:
$$\int_{f(x)}^{g(x)} \phi (t)dt=\phi [g(x)]g'(x) - \phi [f(x)]f'(x)$$

5. Jan 14, 2004

himanshu121

Oh Yes Thanks Halls for correcting