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Math Help - Double integral problem

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    Double integral problem

    So in order to evaluate this double integral, I first evaluate the inner part with regards to x and consider y a constant. So far so good, but I'm not sure how to find the anti-derivative for this expression. I tried rewriting it to make it simpler. I know that to find the antiderivative of ln (root)x I have to use integration by parts, but I'm not sure what to do when there's an additional kx^-1. Any help would be appreciated!
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    Quote Originally Posted by gralla55 View Post
    So in order to evaluate this double integral, I first evaluate the inner part with regards to x and consider y a constant. So far so good, but I'm not sure how to find the anti-derivative for this expression. I tried rewriting it to make it simpler. I know that to find the antiderivative of ln (root)x I have to use integration by parts, but I'm not sure what to do when there's an additional kx^-1. Any help would be appreciated!

    \ln\sqrt{x}=\frac{1}{2}\ln x , so the first integral you want is \frac{1}{2}\int\ln x\cdot \frac{1}{x}\,dx , which is of the almost

    immediate form \int f(x)f'(x)dx=\frac{f(x)^2}{2} + C

    Tonio
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    Thank you so much! The rest of the problem was easy once that was figured out.

    I'm stuck with yet another double integral. Any hints on what to do next? Thanks again!
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    Quote Originally Posted by gralla55 View Post
    Thank you so much! The rest of the problem was easy once that was figured out.

    I'm stuck with yet another double integral. Any hints on what to do next? Thanks again!

    You can't integrate e^{-x^2} automatically like that: this function has no primitive expressable in elementary functions.

    You better change the integration order using Fubbini, and get:

    \int\limits_0^1\int\limits_0^{2x}e^{-x^2}dydx=\int\limits_0^1 2xe^{-x^2}dx=-e^{-x^2}|_0^1 = 1-\frac{1}{e}

    Tonio
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