To prove: The mean value theorem for the integral function .
The mean value theorem of integrals is .
Show the part 2 of fundamental theorem of calculus.
The function is .
Show the expression for mean value theorem of integrals.
Applying the condition for mean value theorem for integrals.
There is a exists number of c. Hence, the expression of function as follows:
Apply mean value theorem for derivative.
Replace for in Equation (1).
Apply part 2 of fundamental theorem of calculus.
Thus, The mean value theorem of integrals is .
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