Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int 1/(t^5) dt` from `[1,x]`?

Answer 1

You could either evaluate the integral and then differentiate or reason as follows:

Explanation:

The Second Fundamental Theorem of Calculus says that

If `f` is continuous on `[a,b]`, then

`int_a^b f(x) dx = F(b)-F(a)`
where `F` is a function for which `F'(x) = f(x)` for all `x` in `[a,b]`.

In this case we are using the variable `t` in the integrand and the variable `x` as the upper limit of integration.

We want the derivative of `int_1^x 1/t^5 dt`.

Note that since we are asked about the interval `[1,x]`, we must have `x > 1` (otherwise the interval is either empty or undefined).

So, `1/t^5` is continuous on `[1,x]`, and

`int_1^x 1/t^5 dt= F(x) - F(1)` where `F` is a function such that `F'(x) = 1/x^5`.

And there is our answer. The derivative of `int_1^x 1/t^5 dt` is `1/x^5`.