Question

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]`?

Answer 1

For `f` as stated, the derivative is: `f'(x) = -sqrt(4+sec(x))`

Explanation:

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

`f(x) = int_a^x g(t) dt` for `x` in some interval `[a,b]` on which `g` is continuous.

`f'(x) = g(x)`

Your question asks about the function defined by integrating from `x` tp `pi`:

`f(x)=int_x^pi sqrt(4+sec(t))dt`

so we first need to reverse the order of integration to go from a constant (in this case `pi`) to the variable, `x`. Reversing the order of integration simply introduces a negative sign.

`f(x)=-int_pi^x sqrt(4+sec(t))dt`,

so `f'(x) = -sqrt(4+sec(x))`

If the intended function is

`f(x)=int_pi^x sqrt(4+sec(t))dt`, the omit the reversing of limits of integration and we get `f'(x) = sqrt(4+sec(x))`