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Math Help - FTC Question

  1. #1
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    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.
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  2. #2
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    Re: FTC Question

    Quote Originally Posted by cpbrunner View Post
    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))
    Thanks from topsquark and cpbrunner
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  3. #3
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    Re: FTC Question

    CP....
    The answer is 2x\sqrt{1-x^2}
    have a look here:
    The Fundamental Theorem of Calculus
    Thanks from cpbrunner
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  4. #4
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    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.FTC Question-15-jun-13.png
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  5. #5
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    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......
    Thanks from ibdutt
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  6. #6
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    Re: FTC Question

    Thanks you are right missed out on negative sign for derivative of - x^2 should have been -2x . Thanks
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  7. #7
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    Re: FTC Question

    Thanks for the help everyone!
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