1. ## FTC Question

I found this problem in a textbook, and it seems interesting, but I'm having trouble solving it.

Evaluate this integral twice: $\frac{d}{dx}\int_{-5}^{x^2}\sqrt{1-t}\,dt$ Once, using FTC 1, and again, using FTC 2.

I've got the FTC 1 part (integrate, then differentiate), which would give me $2x\sqrt{1-x}$. Does anyone have any ideas or help for the second part? I'm stuck on how to apply the FTC.

2. ## Re: FTC Question

Originally Posted by cpbrunner
I found this problem in a textbook, and it seems interesting, but I'm having trouble solving it.

Evaluate this integral twice: $\frac{d}{dx}\int_{-5}^{x^2}\sqrt{1-t}\,dt$ Once, using FTC 1, and again, using FTC 2.

I've got the FTC 1 part (integrate, then differentiate), which would give me $2x\sqrt{1-x}$. Does anyone have any ideas or help for the second part? I'm stuck on how to apply the FTC.
If each of $f~\&~g$ is a diferentiable function then $\frac{d}{{dx}}\int_{g(x)}^{f(x)} {h(t)dt} = f'(x)h(f(x)) - g'(x)h(g(x))$

3. ## Re: FTC Question

CP....
have a look here:
The Fundamental Theorem of Calculus

4. ## Re: FTC Question

Minoanman has given a very good reference. i feel to understand the basic steps you may also have a look at the attachment.

5. ## Re: FTC Question

Ibdutt

check your solution .it is wrong... get out the minus sign....
the correct solution is 2xsqrt(1-x^2).
Sorry I cannot use the Latex......

6. ## Re: FTC Question

Thanks you are right missed out on negative sign for derivative of - x^2 should have been -2x . Thanks

7. ## Re: FTC Question

Thanks for the help everyone!