Fundamental Theorem of Calculus

Theorem I

If is continuous on that is differentiable on and if is integrable on then

Proof Hint

Consider a general partition and use Mean Value Theorem on each parition.

Integration by parts

Suppose are continuous functions on that are differentiable on . If and are Riemann integrable on :

Proof Hint

Consider and use FTC I.

Theorem II

Suppose is an Riemann integrable function on . For .

• is uniformly continuous on
• is continuous at is differentiable and

Proof Hint

For the first point:

• Consider 2 points in the interval such that
• Show

For the second point: Consider the continuity definition of and prove is quite trivial.

Theorem

Suppose is Riemann integrable on an open interval containing the values of differentiable functions . Then:

Proof Hint

Can be done using FTC I and II. Proof is quite trivial.

Theorem - Change of Variable

Suppose is a differentiable function on an open interval such that is continuous. Let be an open interval such that .

If is continuous on , then is continuous on and: