$$\begin{align*} & \int_{0}^{\frac{\pi }{2}}{{{\ln }^{n}}\sin x\text{d}x} \\ & \int_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\ln \left( \ln \tan x \right)}\text{d}x \\ \end{align*}$$
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2The $2$nd integral is called Vardi's integral and you may find here a solution: http://math.stackexchange.com/questions/285671/vardis-integral-int-pi-4-pi-2-ln-ln-tan-xdx. For the first integral I wonder if it helps to let $\ln\sin x = u$. – user 1591719 Jan 27 '13 at 08:48
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@Chris'ssister :Thx Chris : ) – Ryan Jan 27 '13 at 09:01
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nice questions +1 – user 1591719 Jan 27 '13 at 09:01
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1@Chris'ssister: thanks for the link. – Ron Gordon Jan 27 '13 at 10:58
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@rlgordonma: welcome :-) – user 1591719 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 $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$
