Second Fundamental Theorem

To understand what this Theorem is saying, do the following:

• Integrate f(t) = t^2 + 1 from 0 to x. What expression do you get? Call this F(x)
• Take the derivative of this expression with respect to x. What do you get?

Is your derivative equal to f(x)?

The Second Fundamental Theorem says that when any function f(t) is integrated from 0 to x to get F(x), the derivative of this function F(x) is f(x).

Whatever f(t) is we can define F(x) to be the integral from 0 to x of f(t).

• The graphical interpretation is that F(x) is the total area beneath the f(t) curve from t = 0 to t = x.  Your text contains many examples of this sort of graphical interpretation of the integral.  You should be able to construct the graph of F(x) from the graph of f(t).
• If f(t) is a function you can integrate you can find F(x), as in the above example.  Then when you take the derivative of F(x) you get f(x).
• A very simple example:  when you integrate f(t) = t from 0 to x you get F(x) = x^2 / 2; the derivative of F(x) is just F ' (x) = x, so f(x) = F ' (x).  The theorem just says that the derivative of the antiderivative is the original function.   No surprise.

So if f(t) = Si ( ln ( J(t) ), where J(t) is some mysterious Bessel function, we can define F(x) to be the integral from 0 to x of this function.

• You can't write down the antiderivative you need to integrate that function.   I don't think I can.  In fact I don't think anyone can.
• However, over appropriate domain the function is continuous so it can be integrated--whether we can write it down or not the integral exists.
• If F(x) is the integral, we still know F ' (x).  Just as in the previous examples F ' (x) = f(x).
• So F ' (x) = Si (ln (J(x) ) ).

You should pick several functions for f(t)--functions you can integrate--and integrate each from 0 to x. 

• You should then take the derivative of each integral and compare with the original function. 

You should also graph your original function and construct the graph of the integral from 0 to x.

• Don't plot the function you got when you integrated, actually construct it (you do this by making the slope of the integral function equal to the value of the original function).
• Then see if your graph looks like that of the function you got when you integrated.