The Second Fundamental Theorem of Calculus The Mean Value and Average Value Theorem For Integrals
Example The average value of y = sin x between x = 0 and
x = p is
The Second Fundamental Theorem of Calculus
Eample
Example:
Solution This is not in the form where second fundamental theorem
of calculus can be applied because of the x^{2}. We use the chain
rule so that we can apply the second fundamental theorem of calculus. Example Find the derivative of
Solution Here, the "x" appears on both limits. We use two properties of integrals to write this integral as a difference of two integrals.
The first integral can now be differentiated using the second fundamental theorem of calculus,
The second integral can be differentiated using the chain rule as in the last example.
so that
Putting these results together gives the derivative of 2 sin(4x^{2})  sin(x^{2}) Click here for a challenge problem 
