Indefinite Integration 4.1, 4.4, 4.5 Evita Miller, Alli Neal, Perris Howard, & Kelly Keffeler

Power Rule for Integration

Fundamental Theorems of Calculus The Fundamental Theorem of Calculus, which states: If f is continuous on the interval [a,b] then . In other words, the definite integral of a derivative gets us back to the original function. What if we instead change the order and take the derivative of a definite integral? If the limits are constant, the definite integral evaluates to a constant, and the derivative of a constant is zero, so that's not too interesting. If the definite integral represents an accumulation function, then we find what is sometimes referred to as the Second Fundamental Theorem of Calculus: . In other words, the derivative of a simple accumulation function gets us back to the integrand, with just a change of variables (recall that we use t in the integral to distinguish it from the x in the limit).

Fundamental Theorems Simplified 1st Fundamental Theorem- +C’s cancel out in definite integration. 2nd Fundamental Theorem- The derivative of an integral equals the function. They cancel each other out and you just plug in the variable.

Mean Value Theorem Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that

Mean Value Theorem of Integration If f is continuous on [a,b] then there exists a point in [a,b] such that f(c) is the average value of the function in [a,b] Average Value= Height width f(c)

Helpful Sites http://www.bbc.co.uk/scotland/learning/bitesize/higher/maths/calculus/integration_rev1.shtml

The Chain Rule Example:

Anti-Differentiation of Composite Functions or Example:

Constant Multiple Rule Example:

Substitution Method Example: let

General Power Rule for Integration Example: