Integral Calculus: Fund Theorem Confusion

[Alem2000]

Hello page...im taking integral calculus and we are past integration of improper integrals. I know how to use the fundamental theorem but i don't 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...please

 

[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.

 

[Alem2000]

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


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

 

[mathwonk]

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

 

[Tide]

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

 

[mathwonk]

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

 

[JasonRox]

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

 

[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.