The average value of y = sin x between x = 0 and
x = p is
This is not in the form where second fundamental theorem
of calculus can be applied because of the x
Find the derivative of
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
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