# Functions defined by integrals.

Theorem 8.8.1 (The Fundamental Theorem of Calculus)   Let be a function, continuous on the interval . For , we define the function as follows:

.

Then is differentiable on and .

With another notation:

Example 8.8.2

Let . We prove that is differentiable on and we will compute the first derivative of .

As the sine function is continuous on , the function is well-defined on . For any real , we have:

The function is differentiable on , by Theorem 5.9 and Theorem 8.1. We have:

The function is differentiable on , by Theorem 8.1; we have:

Thus is differentiable on and

Actually, in Example 8.2, we could have computed an explicit formula for . In the Example 8.3, this should be impossible.

Example 8.8.3

Let . The function defined by is continuous on , therefore by Theorem 8.1, the function is differentiable on and we have:

Noah Dana-Picard 2007-12-28