I need some advice on how to evaluate it. $$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $$ Thanks!
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1The integral is not well-defined. $\ln(1/e)=-1$ ... – David Mitra May 22 '13 at 17:02
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Try a u-substitution with $u=\ln(x)$. – Andy Bromberg May 22 '13 at 17:03
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so I need to change $\int\limits_\frac{1}{e}^1$ to? – Ofir Attia May 22 '13 at 17:04
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@AndyBromberg if I`m doing it so I get $$\int \frac{dt}{\sqrt{t}}$$ but still I need to change the values of the integral right? – Ofir Attia May 22 '13 at 17:06
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Yep! So you'll get: $$\int_{-1}^{0} \frac{dt}{\sqrt{t}}$$ – Andy Bromberg May 22 '13 at 17:08
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@OfirAttia: your solution should look like $2 \sqrt{\log x},\Big|_{\frac{1}{e}}^{1}$ – Alex May 22 '13 at 17:09
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@Alex but ln of $\frac{1}{e}$ is negative and then its not defined there – Ofir Attia May 22 '13 at 17:10
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@OfirAttia, yes, with these bounds you're going to get an imaginary answer. – Andy Bromberg May 22 '13 at 17:11
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@OfirAttia: are you familiar with complex numbers? – Alex May 22 '13 at 17:11
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2@DonAntonio: perhaps you mean $$\int_{1/e}^1\frac1{x\sqrt{|\log(x)|}}\mathrm{d}x$$ – robjohn May 22 '13 at 17:17
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Indeed @robjohn, thanks. – DonAntonio May 22 '13 at 17:19
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@Alex familiar yes but, I just need to say if its improper or not. – Ofir Attia May 22 '13 at 17:21
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A related problem. – Mhenni Benghorbal May 22 '13 at 17:33
2 Answers
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Here's a hint: $$ \int_{1/e}^1 \frac{1}{\sqrt{\ln x}} {\huge(}\frac{dx}{x}{\huge)}. $$ What that is hinting at is what you need to learn in order to understand substitutions. It's all about the chain rule. The part in the gigantic parentheses becomes $du$.
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I understand, but still my problem is the know when to change the limits, I get : $$\int^1_\frac{1}{e} \frac{dt}{\sqrt{t}}$$ – Ofir Attia May 22 '13 at 17:23
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You have $t = \ln x$. When $x = 1/e$, then $t=\ln(1/e)=-1$. When $x=1$, then $t=\ln 1 = 0$. So you have $\displaystyle\int_{-1}^0 \frac{dt}{\sqrt{t}}$. Since you're talking about square roots of negative numbers, you have a question of how to make sense of those. One branch, maybe. – Michael Hardy May 22 '13 at 20:54
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To recap all that happened in the comments section:
Based on the initial problem of$$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}}$$
We perform a u-substitution with $u=\ln{x}$ and $du=\frac{dx}{x}$. Also, the bounds are converted to $\ln\frac{1}{e}=-1$ and $\ln{1}=0$. So we have:
$$\int\limits_{-1}^{0} \frac{du}{\sqrt{u}}=2\sqrt{u}\big|_{-1}^{0}=0-2i=-2i$$
And that's the solution!
Andy Bromberg
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