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Math Help - Integral of ln (x)

  1. #1
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    Integral of ln (x)

    First, I must disclose that this is a homework problem.

    Here is the integral exactly as stated:

    Evaluate

    <br />
\int\limits_0^1 {\ln x} dx<br />

    So this is what I did:

    <br />
\int\limits_0^1 {\ln x} dx = x\ln x - x|_0^1 <br />

    However, ln 0 is undefined. So how do I proceed? Or, is it possible to proceed?

    Any help is greatly appreciated.
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  2. #2
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    Quote Originally Posted by kid funky fried View Post
    First, I must disclose that this is a homework problem.

    Here is the integral exactly as stated:

    Evaluate

    <br />
\int\limits_0^1 {\ln x} dx<br />

    So this is what I did:

    <br />
\int\limits_0^1 {\ln x} dx = x\ln x - x|_0^1 <br />

    However, ln 0 is undefined. So how do I proceed? Or, is it possible to proceed?

    Any help is greatly appreciated.
    No, This is called an "improper integral", It should be solved in terms of limits.
    \int_0^1 ln(x) dx=\lim_{t \to 0^+} \int_t^1 ln(x) dx.
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  3. #3
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    Quote Originally Posted by kid funky fried View Post
    First, I must disclose that this is a homework problem.

    Here is the integral exactly as stated:

    Evaluate

    <br />
\int\limits_0^1 {\ln x} dx<br />

    So this is what I did:

    <br />
\int\limits_0^1 {\ln x} dx = x\ln x - x|_0^1 <br />

    However, ln 0 is undefined. So how do I proceed? Or, is it possible to proceed?

    Any help is greatly appreciated.
    This is an improper integral.

    \int_0^1ln(x)dx=\lim_{t->0^+}\int_t^1ln(x)dx

    Replace the 0 by t in your equation you obtained by IBP and then calculate this as a limit.
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