Question

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

Answer 1

`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}`