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$$\begin{align*} & \int_{0}^{\frac{\pi }{2}}{{{\ln }^{n}}\sin x\text{d}x} \\ & \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\ln \left( \ln \tan x \right)}\text{d}x \\ \end{align*}$$

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    $\begingroup$ The $2$nd integral is called Vardi's integral and you may find here a solution: math.stackexchange.com/questions/285671/…. For the first integral I wonder if it helps to let $\ln\sin x = u$. $\endgroup$ – user 1357113 Jan 27 '13 at 8:48
  • $\begingroup$ @Chris'ssister :Thx Chris : ) $\endgroup$ – Ryan Jan 27 '13 at 9:01
  • $\begingroup$ nice questions +1 $\endgroup$ – user 1357113 Jan 27 '13 at 9:01
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    $\begingroup$ @Chris'ssister: thanks for the link. $\endgroup$ – Ron Gordon Jan 27 '13 at 10:58
  • $\begingroup$ @rlgordonma: welcome :-) $\endgroup$ – user 1357113 Jan 27 '13 at 14:33
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The first integral is the log-sine integral. See this post to see how to evaluate it: Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$

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