Rieman Integral: The Fundamental Theorem of Calculus

[kingstrick]

Homework Statement

Let I := [a,b] and let f: I→ℝ be continuous on I. Also let J := [c,d] and let u: J→ℝ be differentiable on J and satisfy u(J) contained in I. Show that if G: J→ℝ is defined by

G(x) :=∫^u(x)_af for x in J, then G'(x) = (f o u)(x)u'(x) for all x in J.


2. The attempt at a solution

I am not sure what theorem to use in this situation. I am thinking that the product and composition of differentiable functions means that G'(x) exists...but I don't believe that will help me in this situation. Any ideas as to how i should start this one?

 

[Dick]

Written a little better the integral is $\int^{u(x)}_a f(t) dt$. Now use http://en.wikipedia.org/wiki/Leibniz_integral_rule

 

[HallsofIvy]

The Leibniz formula that Dick links to is basically using the chain rule.

If $F(x)=\int_a^{g(x)} f(t)dt$ let u= g(x) so that the integral is
$\int_a^u f(t)dt$ and, by the Fundamental theorem of Calculus,
$$\frac{d}{du}\int_a^u f(t)dt= f(u)$$

Now, use the chain rule to find the derivative with respect to x.

 

[kingstrick]

I don't understand. Isn't the derivative of G(x) the f in the integral. Or am i supposed to find the derivative of F(X)?

 

[HallsofIvy]

No, the derivative of G(x) is NOT "the f in the integral". If the upper limit were just "x" it would be.