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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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The $2$nd integral is called Vardi's integral and you may find here a solution:…. For the first integral I wonder if it helps to let $\ln\sin x = u$. – user 1618033 Jan 27 '13 at 8:48
@Chris'ssister :Thx Chris : ) – Ryan Jan 27 '13 at 9:01
nice questions +1 – user 1618033 Jan 27 '13 at 9:01
@Chris'ssister: thanks for the link. – Ron Gordon Jan 27 '13 at 10:58
@rlgordonma: welcome :-) – user 1618033 Jan 27 '13 at 14:33
up vote 2 down vote accepted

The first integral is the log-sine integral. See this post to see how to evaluate it: Solve the integral $\displaystyle{S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx}$

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