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Math Help - log and integrals

  1. #1
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    Question log and integrals

    Hello

    In this problem, need to find the area below the function:
    1.
    f(x) = \frac{2-2x}{1+2x} in the interval [0,1].

    And above the x axis.

    I am told there is some logarithms involved here.


    After sitting all night....another problem:

    2.
    Prove that the function has an absolute minimum.

     f(x) = x^2+(\ln{x})^2

    Thanks.
    Last edited by ryu1; May 24th 2013 at 06:53 PM.
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  2. #2
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    Re: log and integrals

    Hey ryu1.

    The area will involve the integral so you need to be able to solve the integral.

    Hint: Try substituting in (2 - 1 + 1) and getting your integral in terms -1 + a/(1+2x) dx and consider the derivative of log(1+2x).
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  3. #3
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    Re: log and integrals

    Quote Originally Posted by ryu1 View Post
    Hello

    2.
    Prove that the function has an absolute minimum.

     f(x) = x^2+(\ln{x})^2

    Thanks.
    As for this problem, can I just say that lnx is continuous for all X>0
    and
    \lim_ {x \to 0^+}{x^2+(lnx)^2} = +\infty

    and also
    \lim_ {x \to +\infty}{x^2+(lnx)^2} = +\infty

    therefore an absolute minimum is guaranteed ? Thanks.
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  4. #4
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    Thumbs up Re: log and integrals

    Quote Originally Posted by chiro View Post
    Hey ryu1.

    The area will involve the integral so you need to be able to solve the integral.

    Hint: Try substituting in (2 - 1 + 1) and getting your integral in terms -1 + a/(1+2x) dx and consider the derivative of log(1+2x).

    Got it!

    Thanks, problem solved
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