Definition of derivative integration problem

[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).

 

[Hurkyl]

Try going back to the definition of derivative.

 

[himanshu121]

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

 

[HallsofIvy]

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

 

[himanshu121]

Oh Yes Thanks Halls for correcting