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Math Help - Integral with limit

  1. #1
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    Integral with limit

    Express the limit as a definite integral on the given interval.
    lim n
    n xi ln(1 + xi2) Δx, [0, 4]
    i = 1





    My work: I figured out that f(x) = xln(1+x^2)

    therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx



    I'm a little confused. Do I start by making a graph? I haven't memorized the log graphs so sorry for my naivety. I appreciate any help.
    Last edited by Steelers72; February 15th 2013 at 12:47 PM.
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  2. #2
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    Re: Integral with limit

    Quote Originally Posted by Steelers72 View Post
    Express the limit as a definite integral on the given interval.
    lim n
    n xi ln(1 + xi2) Δx, [0, 4]
    i = 1





    My work: I figured out that f(x) = xln(1+x^2)

    therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx

    Your work is correct. What more do you need to do?
    It does not ask you to evaluate it. Does it?
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  3. #3
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    Re: Integral with limit

    Quote Originally Posted by Steelers72 View Post
    Express the limit as a definite integral on the given interval.
    lim n
    n xi ln(1 + xi2) Δx, [0, 4]
    i = 1





    My work: I figured out that f(x) = xln(1+x^2)

    therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx



    I'm a little confused. Do I start by making a graph? I haven't memorized the log graphs so sorry for my naivety. I appreciate any help.
    If you need to evaluate the integral \displaystyle \begin{align*} \int_0^4{x\ln{\left( 1 + x^2 \right)} \, dx } = \frac{1}{2} \int_0^4{ 2x\ln{ \left( 1 + x^2 \right) } \,dx } \end{align*}, let \displaystyle \begin{align*} u = 1 + x^2 \implies du = 2x\, dx \end{align*} and note that \displaystyle \begin{align*} u(0) = 1 \end{align*} and \displaystyle \begin{align*} u(4) = 17 \end{align*}, then

    \displaystyle \begin{align*} \frac{1}{2}\int_0^4{2x\ln{\left( 1 + x^2 \right) }\, dx} &= \frac{1}{2} \int_1^{17} { \ln{(u)} \, du } \end{align*}

    which can now be solved by integration by parts.
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    Re: Integral with limit

    Thank you both; the work I had done was all we needed to do haha...sorry for that!
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