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

1 Answer
Apr 11, 2016

Answer:

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).