Derivative of an anti-derivative?

Question

Problem (int means the integral sign):

d/dx( int(arccos(t^7)dt) from 2 to x )

the answer is arccos(x^7) but can you explain the steps? Is the fundamental theorem of calculus involved? This was very poorly explained in my book and I wanted to see if any of you have insight...

Answer 1

Well, let's do the operations step by step first and see what happens

First lets do int(arccos(t^7)dt) from 2 to x

Let f(t) = arccos(t^7)

and let F(t) exist such that dF(t)/dt = f(t)

then, int(f(t)dt) from 2 to x = F(x) - F(2) (Fundamental Theorem of Calculus, Part 2)

Then if you take d/dx.

d/dx (F(x) - F(2)) = dF(x)/dx (The second term of the derivative is 0 since F(2) is a constant)

dF(x)/dx = f(x) = arccos(x^7) by earlier definition of the function F.

So therefore, the Fundamental Theorem of Calculus is indeed involved.

Answer 2

The FTC says that the derivative of x ----> int_a^x f(t) dt at x
is f(x) if f is continuous at x. So what else can you possibly need?

Answer 3

shouldn't you worry about your avatar santa hat 1st ???