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

Apr 22, 2018

$F ' \left(x\right) = {\csc}^{2} x$

#### Explanation:

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

$F \left(x\right) = {\int}_{a}^{x} f \left(t\right) \mathrm{dt} ,$ $a$ is any constant, then

$F ' \left(x\right) = f \left(x\right)$

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

$F \left(x\right) = {\int}_{0}^{x} {\csc}^{2} t \mathrm{dt}$, and we see $f \left(t\right) = {\csc}^{2} t , f \left(x\right) = {\csc}^{2} x ,$ so

$F ' \left(x\right) = {\csc}^{2} x$