Fundament Confusion

1. Sep 25, 2004

Alem2000

Hello page...im taking integral calculus and we are past integration of improper integrals. I know how to use the fundamental theorm but i dont get the first part...$\frac{d}{dx}\int^x_af(t)dt=f(x)$

the book used it in an example....find the dirivative of $$g(x)=\int_0^1\sqrt{1+t^2}dt$$....the book goes on to tell you the answer but it show NO STEPS...it is James Stewart Calculus 2nd edition i believe if anyone has the same book..page 383..but anyway can some one go through the steps...plz

Last edited: Sep 25, 2004
2. Sep 25, 2004

mathman

There is something wrong in the statement of the problem. You have g(x) = something, where x does not appear. As stated g'(x)=0.

3. Sep 25, 2004

Alem2000

sorry I copied down the problem wrong...this is the correct one

$$g(x)=\int_0^x\sqrt{1+t^2}dt$$

4. Sep 25, 2004

mathwonk

just plug into the statement you gave of the ftc. this is a special case. i.e. what is f(t) here?

5. Sep 25, 2004

Tide

We can get it from first principles:
$$\frac {dg}{dx} = \lim_{h \rightarrow 0} \frac {\int_0^{x+h} \sqrt{1+t^2} dt - \int_0^{x} \sqrt {1+t^2} dt}{h}$$
$$\frac {dg}{dx} = \lim_{h \rightarrow 0} \frac {\int_{x}^{x+h} \sqrt{1+t^2} dt}{h}$$
$$\frac {dg}{dx} = \lim_{h \rightarrow 0} \sqrt{1+x^2} \frac {\int_{x}^{x+h} dx}{h} = \sqrt {1+x^2}$$

6. Sep 25, 2004

mathwonk

what "first principle" did you use in the next to last step?

7. Sep 26, 2004

JasonRox

I got the fifth edition, so I can't help you there. I just hope I don't encounted similiar problems.

8. Sep 26, 2004

matt grime

what problems do you think the OP encountered in the book? the solution is self evident and any problems the OP had are nothing to do with the book. the book has many faults, if it's the one i think it is, but that isn't one of them.