Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int sect tant dt` from `[0, x^3]`?

Answer 1

You have two choices for how to do this.

Explanation:

Method 1
`int_0^(x^3) sect tant dt = F(x^3)-F(0)` where `F` is an antiderivative of `sect tant`.

Now use the chain rule to differentiate `F(x^3)` with respect to `x`

We get `secx^3 tanx^3 (3x^2)`

Method 2

(Actually evaluate the definite integral.)

Since `d/dt(sect) = sect tant`, we get `sect` is an antiderivative of `sect tant`.

So, applying the second fundamental theorem of calculus, we find

`int_0^(x^3) sect tant dt = sect]_0^(x^3) = secx^3-sec0`

Now we differentiate to answer the question:

`d/dx(sec(x^3)) = secx^3 tanx^3 (3x^2)` (rearrange to taste).