Show steps : ∫ (1/x log(log(x))) dx

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Xaphene: Show steps : ∫ (1/x log(log(x))) dx

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May 26, 2014 at 1:51am

Shu Yun:Is it ∫ ((1/x) log(log(x))) dx or ∫ (1/ (x log(log(x))) ) dx ?
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- May 26, 2014 at 10:15am
Xaphene:first one... ∫ ((1/x) log(log(x))) dx
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- May 26, 2014 at 10:46am
Shu Yun:Assume yr log means log10 (lg). C[deqn]\int(\frac{1}{x}\cdot\lg(\lg x))~dx\\ =\int(\frac{1}{x}\cdot\lg(\frac{\ln x}{\ln 10}))~dx\\ =\int(\frac{1}{x}\cdot\frac{\ln(\frac{\ln x}{\ln 10})}{\ln10})~dx\\ =\frac{1}{\ln10}\int\frac{1}{x}\cdot\ln(\frac{\ln x}{\ln10})~dx\\ =\frac{1}{\ln10}\left[ \ln x\cdot \ln(\frac{\ln x}{\ln10})~-\int\ln x\cdot\frac{\frac{1}{\ln10}(\frac{1}{x})}{\frac{1}{\ln10}(\ln x)}~dx\right] ~(integration~by~parts)\\ =\frac{1}{\ln10}\lceil \ln x\cdot\ln(\frac{\ln x}{\ln10})~-\ln\left| x \right| \rceil+c[/deqn]hange log to ln first
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- May 26, 2014 at 11:25am
Shu Yun:*Change log to ln first
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- May 26, 2014 at 11:25am
Xaphene:no log cannot be changed to natural logarithm (In log). can you explain the steps to fit this answer in the format (x/2)+(1/4)sin(2x)+C
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- May 26, 2014 at 11:44am
Shu Yun:Err really? Coz i can prove it with calculator! Anyway sorry i'm not sure how to answer yr second part though..
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- May 26, 2014 at 11:54am
Xaphene:I got this... ∫(1/x)log(log x) dx Let u = log x Diff. both sides w.r.t. x, we get (du/dx) = (d/dx)(log x) => du = (1/x) dx Let I = ∫ log u du =>I= log u . { ∫ 1 du} - ∫ { (d/du){log (x) . ∫ 1 du} du = u(log u) - ∫ 1 du = u(log u) - u + C = log x{log(log x)}-log x + C ---------------------------------------------------"
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- May 27, 2014 at 6:07am