Derivative of an integral

Let f be a Riemann integrable function on an interval [a,b]. Fix some c from this interval. Let g be a function on [a,b] with values in [a,b]. Then

If the variable x appears at the lower limit of the integral and the upper limit is a constant, we switch the limits using the usual formula (the minus sign appears in front of the integral).
If the variable x appears at both limits of the integral, we split the integral at some constant and differentiate each part separately according to this formula and the trick for variable in lower limit.

Example: For x from (0,π) we define

Find the derivative of F.

Solution:

Gamma and Beta functions
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