Question

How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function `h(x) = 1/x int 2 arctan t dt`?

Answer 1

If the intended function is `h(x) = int_x^1 2arctantdt`, then see below.

Explanation:

Part 1 of the fundamental theorem of calculus tells us how to differentiate functions of the form

`g(x) = int_a^x f(t) dt` for constant `a`

(We get `g'(x) = f(x)`.)

The function `h`, in this question is not quite in the form we need, but we can rewrite it:

`h(x) = - int_1^x 2arctant dt`

So, `h'(x) = -2arctanx`