# FTC Question

• Jun 14th 2013, 10:41 AM
cpbrunner
FTC Question
I found this problem in a textbook, and it seems interesting, but I'm having trouble solving it.

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

I've got the FTC 1 part (integrate, then differentiate), which would give me $\displaystyle 2x\sqrt{1-x}$. Does anyone have any ideas or help for the second part? I'm stuck on how to apply the FTC.
• Jun 14th 2013, 12:01 PM
Plato
Re: FTC Question
Quote:

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

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

If each of $\displaystyle f~\&~g$ is a diferentiable function then $\displaystyle \frac{d}{{dx}}\int_{g(x)}^{f(x)} {h(t)dt} = f'(x)h(f(x)) - g'(x)h(g(x))$
• Jun 14th 2013, 12:31 PM
MINOANMAN
Re: FTC Question
CP....
have a look here:
The Fundamental Theorem of Calculus
• Jun 14th 2013, 08:22 PM
ibdutt
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.Attachment 28619
• Jun 14th 2013, 09:13 PM
MINOANMAN
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......
• Jun 14th 2013, 10:15 PM
ibdutt
Re: FTC Question
Thanks you are right missed out on negative sign for derivative of - x^2 should have been -2x . Thanks
• Jun 16th 2013, 11:29 AM
cpbrunner
Re: FTC Question
Thanks for the help everyone!