On-line Math 21

5.2  The Fundamental Theorem of Calculus

Example 2 Find the derivative of

F(x) = ó õ x2 0  Ö   1+t6   dt.

Solution

Here again, finding the integral would be hard, if not impossible. But that is not what you are asked. If

G(x): = ó õ x 0  Ö   1+t6   dt,

(note that the limits of integration are a little different), then

F(x) = G(x2),

or

F(x) = G(u), where u = x2.

The FTC we just proved shows that

dG du = Ö   1+u6   .

Don't be confused by the use of u instead of x . It's actually supposed to help. If I had used x instead of the u , that would be exactly what the FTC says.

Then, by the chain rule,

dF dx = dG du du dx = Ö   1+u6   2x = Ö   1+(x2)6   2x = 2x Ö   1+x12   .

Copyright (c) 2000 by David L. Johnson.