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Math Help - Fundemental Theorem Questions (I think)

  1. #1
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    Question Fundemental Theorem Questions (I think)

    For maths, i dont know how to work out the following:

    1)Find the value of;
    Integral(0,4): f'(x) ( (f(x))^3 -2f(x) -3) dx
    given that f(0) = 2, f'(0) = -6, f(4) = 3 and f'(4) = -3

    2) y= Integral(5,5x^4):sin( t^2 - 5 ) dt
    whats y'?

    (I'm new so sorry, i dont know how to write in equation form)
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  2. #2
    Super Member wingless's Avatar
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    Hint:

    \int f'(x) [f(x)]^n~dx

    Let u = f(x), then du = f'(x)~dx.

    \int u^n~du

    - If n\neq -1, then it's \frac{u^{n+1}}{n+1}= \frac{[f(x)]^{n+1}}{n+1}.

    - If n = -1, then it's \ln u = \ln f(x)
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  3. #3
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by skirk34 View Post
    For maths, i dont know how to work out the following:

    1)Find the value of;
    Integral(0,4): f'(x) ( (f(x))^3 -2f(x) -3) dx
    given that f(0) = 2, f'(0) = -6, f(4) = 3 and f'(4) = -3
    set u = f(x)
    du = f'(x)dx

    at x=0: u=f(0)=2
    at x=4: u=f(4)=3

    thus

    \int_2^3 (u^3 -2u -3)du = ...
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by skirk34 View Post
    For maths, i dont know how to work out the following:

    2) y= Integral(5,5x^4):sin( t^2 - 5 ) dt
    whats y'?

    (I'm new so sorry, i dont know how to write in equation form)
    if F(x) = \int_a^{f(x)} h(t)dt where a is a constant,

    then  F'(x) = h(f(x)) \, f'(x)
    Last edited by kalagota; August 1st 2008 at 06:09 AM.
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  5. #5
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    For the second question I got the answer:
    y'(x) = sin(x^2 - 5).20x^3

    The quiz its from isn't liking my answer though
    Heres the hint it gives me:
    Let the integrand be f(t) and let an antiderivative of f(t) be F(t). Then y(x) = F(5 x^4) - F(5), where F(5) is just a constant.
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by skirk34 View Post
    For the second question I got the answer:
    y'(x) = sin(x^2 - 5).20x^3

    The quiz its from isn't liking my answer though
    Heres the hint it gives me:
    Let the integrand be f(t) and let an antiderivative of f(t) be F(t). Then y(x) = F(5 x^4) - F(5), where F(5) is just a constant.
    im sorry.. i edited my last post.. take a look at it..
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