Question

How do you use the fundamental theorem of calculus to find F'(x) given `F(x)=int (csc^2t)dt` from [0,x]?

Answer 1

`F'(x)=csc^2x`

Explanation:

The first part of the Fundamental Theorem of calculus tells us that if

`F(x)=int_a^xf(t)dt,` `a` is any constant, then

`F'(x)=f(x)`

This connects the ideas of differentiation and integration, telling us that the derivative of a function consisting of another integrated function is really just the integrated function.

Here, we have

`F(x)=int_0^xcsc^2tdt`, and we see `f(t)=csc^2t, f(x)=csc^2x,` so

`F'(x)=csc^2x`