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

1 Answer
Apr 22, 2018

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