Results 1 to 6 of 6

Thread: prove this integral does not exist

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    prove this integral does not exist

    integral ( 1/x, x, 0,1)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    GAMMA Mathematics
    colby2152's Avatar
    Joined
    Nov 2007
    From
    Alexandria, VA
    Posts
    1,172
    Thanks
    1
    Awards
    1
    Quote Originally Posted by szpengchao View Post
    integral ( 1/x, x, 0,1)
    Is that $\displaystyle \int_0^1 \frac{1}{x} dx$?

    If so, note it will not exist at the bound: $\displaystyle x=0$ because not only is the function not continuous at that point (cannot divide by zero), the resultant integral $\displaystyle \ln(x)$ cannot take values less than or equal to zero.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1
    You can notice that $\displaystyle \int_0^1 \frac{1}{x} dx=\int_1^\infty \frac{1}{x} dx$.
    So we have $\displaystyle \int_1^\infty \frac{1}{x} dx=$ limit when b tends to $\displaystyle \infty$ of $\displaystyle \int_1^b \frac{1}{x}dx =$ limit when b tends to $\displaystyle \infty$ of $\displaystyle ln(x)$ evaluated in $\displaystyle b$ minus $\displaystyle ln(x)$ evaluated in $\displaystyle 1$. Which is equal to limit when b tends to $\displaystyle \infty$ of $\displaystyle ln(x)$ evaluated in $\displaystyle b$, which is $\displaystyle \infty$, thus divergent.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    prove

    how can u get the first step
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14
    It is a simple reciprocal substitution, try it with $\displaystyle x=\frac1u.$
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1
    Ok. I said $\displaystyle \int_0^1 \frac{1}{x} dx=\int_1^\infty \frac{1}{x} dx$, this implies that the graph of $\displaystyle f(x)=\frac{1}{x}$ is symmetric with respect to the graph of $\displaystyle y=x$. It means that if $\displaystyle f(x)=\frac{1}{x}$, then $\displaystyle f^{-1}(x)=f(x)$. In other words, f has its own bijection equals to itself.
    Now to prove it, you just have to show that $\displaystyle f(f^{-1}(x))=x$.

    $\displaystyle f(f^1(x))$ is equals to $\displaystyle \frac{1}{f^1(x)}=\frac{\frac{1}{1}}{\frac{1}{x}}=x$. It is demonstrated. If we take a precise example maybe you'll understand better. If $\displaystyle x=5$, then $\displaystyle f(5)=\frac{1}{5}$. That means that to get 5 with our function, we need to evaluated it in $\displaystyle \frac{1}{5}$, which works. Maybe you will understand even better seeing the graph. It's trivial that the area under the curve of $\displaystyle f(x)=\frac{1}{x}$ from $\displaystyle 0$ to $\displaystyle 1$ is equal to the area from 1 to $\displaystyle \infty$. To be rigorous, just say f is its own bijection (and show it as I did for you), that should be enough.
    Last edited by arbolis; May 24th 2008 at 12:17 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove lim(sin(1/x)) as x tends to 0 does not exist
    Posted in the Differential Geometry Forum
    Replies: 10
    Last Post: Jun 15th 2011, 02:28 AM
  2. How to prove the limit does not exist at 0
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Mar 28th 2011, 12:40 PM
  3. prove exist uniformly
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Apr 27th 2010, 10:00 AM
  4. How do you prove that a limit does not exist?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Nov 4th 2009, 06:58 PM
  5. Prove that lim (x -> 0) (1/x^2) does not exist
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 26th 2009, 07:46 AM

Search Tags


/mathhelpforum @mathhelpforum