# How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function #f(x)=int sqrt(4+sec(t))dt# from #[x, pi]#?

For

We can use the fundamental theorem of calculus to find the derivative of a function of the form:

If the intended function is

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To use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function ( f(x) = \int_{x}^{\pi} \sqrt{4 + \sec(t)} , dt ), first, we define a new function, say ( F(x) ), as the integral of ( f(x) ) from a constant lower limit to ( x ):

[ F(x) = \int_{a}^{x} f(t) , dt ]

Then, according to Part 1 of the Fundamental Theorem of Calculus, the derivative of ( F(x) ) with respect to ( x ) is equal to the integrand function ( f(x) ):

[ F'(x) = f(x) ]

So, to find the derivative of ( f(x) ), we just need to evaluate ( F'(x) ). In this case, ( F(x) = \int_{a}^{x} \sqrt{4 + \sec(t)} , dt ), with ( a ) being a constant lower limit.

Once we have ( F(x) ), we can find its derivative ( F'(x) ) using the chain rule and then substitute ( x = \pi ) into ( F'(x) ) to obtain the derivative of ( f(x) ) at ( x = \pi ).

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