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Fundamental Theorem of Calculus

Theorem I

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

Integration by parts

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

Theorem II

Suppose is an Riemann integrable function on . For .

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

Theorem

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

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: