The fundamental theorem of calculus

[CantorSet]

Can we apply the fundamental theorem of calculus to an integrand that's a function of the differentiating variable?

For example, can one still use the fundamental theorem of calculus when trying to find the derivative of this function:

$$f(x) = \int_{0}^{x} \frac{cos(xt)}{t}dt$$

 

[Hurkyl]

Nope. But the idea of the proof can still be used -- you can derive a general formula for
$$\int_{f(x)}^{g(x)} h(x, t) \, dt$$
with suitable conditions on the three functions.

 

[CantorSet]

Nope. But the idea of the proof can still be used -- you can derive a general formula for
$$\int_{f(x)}^{g(x)} h(x, t) \, dt$$
with suitable conditions on the three functions.

Well, here's my guest for the formula:

If

$$H(x) = \int_{f(x)}^{g(x)} h(x, t) \, dt$$,

then

$$H'(x) = h(x,g(x)) - h(x,f(x))$$

 

[Hurkyl]

Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?

 

[CantorSet]

Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?

Let

$$F(x) = \int_{a}^{g(x)} f(x,t) dt$$Now define

$$I(x,y) = \int f(x,t) dt$$

with t = y after indefinite integration over t.

Then

$$F'(x) = I_x(x,g(x)) - I_x(x,a) + I_x(x,g(x))g'(x)$$