# The fundamental theorem of calculus

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
Staff Emeritus
Gold Member
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.

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

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

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