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Math Help - integral problmes

  1. #1
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    integral problmes

    1. Find (f^-1)'(0) if:
    integral sign x, 0 1+sin(sint)dt.
    2. Find a function g such that:
    integral sign x, 0 tg(t)dt= x+x^2
    3. Find all continous functions f satisfying:
    integral sign x,0 f=(f(x))^2+C. for some constant C.
    4. Find F'(x) if F(x)= integral x,0 xf(t)dt. (The answer is not xf(x); perform obvious manipulatipon on the integral before trying to find F'.)
    5. The limit lim integral sign N, a f, if it exists, is denoted by integral sign
    N-> alpha
    integral sign (alpha, a) f

    a) Determine integral sign alpha, 1 x^r dx, if r<-1
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  2. #2
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    Quote Originally Posted by Swamifez View Post
    1. Find (f^-1)'(0) if:
    integral sign x, 0 1+sin(sint)dt.
    This function is not invertible .
    2. Find a function g such that:
    integral sign x, 0 tg(t)dt= x+x^2
    Let, x>0
    f(x)=INTEGRAL (from 0 to x) t*g(t) dt
    We are told that,
    f(x)=x+x^2

    Take the derivative of both sides (apply FT of Calculus).
    f'(x)=x*g(x)
    f'(x)=1+2x
    Thus,
    1+2x=x*g(x)
    Thus,
    1/x+2=g(x)
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  3. #3
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    Quote Originally Posted by Swamifez View Post
    3. Find all continous functions f satisfying:
    integral sign x,0 f=(f(x))^2+C. for some constant C.
    I think it is more appropirate to say, "all differenciable functions".
    ---
    INTEGRAL (from 0 to x) f(t)dt=(f(x))^2+C
    Take derivative of both sides (apply FT of Calculus),
    f(x)=2f'(x)f(x)
    One solution is when,
    f(x)=0 everywhere.
    Otherwise we can divide by it (actually we cannot because there might be a zero point, but that will not happen because the function is differenciable)
    1=2f'(x)
    Thus,
    1/2=f'(x)
    Integrate,
    1/2x+C=f(x)
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