How do you use the Fundamental Theorem of Calculus to find the derivative of #int {1} / {1+t^{2}} dt# from x to 5?

1 Answer
Oct 27, 2017

Answer:

# d/dx \ int_x^5 1/ {1+t^2} \ dt = - 1/ {1+x^2} #

Explanation:

If asked to find the derivative of an integral then you should not evaluate the integral but instead use the fundamental theorem of Calculus.

The FTOC tells us that:

# d/dx \ int_a^x \ f(t) \ dt = f(x) # for any constant #a#

(ie the derivative of an integral gives us the original function back).

We are asked to find:

# E = d/dx \ int_x^5 1/ {1+t^2} \ dt #

(notice the upper bounds of the first integral are not in the correct format for the FTOC to be applied, directly). We can manipulate the definite integral using integral properties:

# E = d/dx \ -int_5^x 1/ {1+t^2} \ dt #
# \ \ = - d/dx \ int_5^x 1/ {1+t^2} \ dt #

We can now apply the FTOC to get;

# E = - 1/ {1+x^2} #