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I am supposed to solve a definite integral which involves $\ln$ and $e$. Nowhere in my textbook can I even find examples of how this would be done.

I do know that $\ln(e(x))$ and $e(\ln(x))$ evaluates to $x$, but I don't know how this is supposed to help me with this question.

Determine the possible value of the definite integral $\int_{e}^{e^2} \frac{\ln(\ln(t))}{t\ \log_{2}(w)}dt$

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Are you sure about the $w$ in the integral ? If it is a constant then you can move it outside the integral. If it is a function of $t$ then please give details. – vanna Sep 10 '12 at 11:31
Hint: Let $u = \ln t$, then $du = \frac{dt}t$ ... – martini Sep 10 '12 at 11:31
@vanna Yes w is a constant. – Michael Frey Sep 10 '12 at 11:32
@MichaelFrey then take $\log(w)$ out of the integral. You are left with something of the form $u'(v)\cdot v'$. Do you not recognise this? – user39572 Sep 10 '12 at 12:04
@JulienGodawatta Ok. I get the taking out of $log(w)$, but how do you see that I am left with something in the form of $u'(v)\cdot v'$? – Michael Frey Sep 10 '12 at 12:25
up vote 3 down vote accepted

Let $u = ln(t)$. Then $du = \frac{dt}{t}$. Let $c = \frac{1}{log_{2}(w)}$

Then $\int_{e}^{e^2} \frac{\ln(\ln(t))}{t\ \log_{2}(w)}dt = c\int_{1}^{2} \ln(u)du = c[u \ln(u) - u]_{1}^{2} = c(2\ln(2) - 1)$

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