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##\int_{\pi/4}^{\pi/2} \log \log \tan x \, dx = \frac{\pi}{2} \log \left( \frac{\sqrt{2\pi}\, \Gamma(3/4)}{\Gamma(1/4)} \right)##

requires knowledge of analytic number theory, specifically L-functions. The paper by Ilan Vardi

This technique was given the systematic analysis in Luis A. Medina and Victor H. Moll's paper

Another technique that doesn't get much coverage is the "method of brackets" by Gonzalez and Moss. I only have the arxiv paper: arXiv:0812.3356v1.

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Complex analysis is beautiful.Does anyone know of any integration techniques that aren't covered in calculus 1/2? For example, today I learned the Weierstrass Substitution. Are there other useful techniques?

The integral ##\displaystyle \int_{0}^{\infty}\frac{1}{x^4+1} \ dx## can be solved by partial fraction decomposition, as you might know from this thread. Or, we can use something called Cauchy's Residue Theorem. A lot of definite integrals become a lot easier if we do them indirectly with contour integrals in the complex plane.

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How would you do that integral with complex analysis?

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To solve the integral with complex analysis, one would have to use a line integral in the complex plane and use the Residue Theorem, as Mandelbroth mentioned.How would you do that integral with complex analysis?

[itex]\displaystyle \int_{0}^{\infty}\frac{1}{x^4+1} \ dx[/itex]

We could start by associating with the given real integral a related contour integral, of the form [itex]\displaystyle \int_{\Gamma}f(z) \ dz[/itex].

We observe that [itex]2\displaystyle \int_{0}^{R}\frac{1}{x^4+1} \ dx = \displaystyle \int_{-R}^{R}\frac{1}{x^4+1} \ dx[/itex].

So, we consider [itex]\displaystyle \int_{\Gamma}\frac{1}{z^4+1} \ dz[/itex], where [itex]\Gamma = [-R,R] \cup \Gamma(R)[/itex], with [itex]\Gamma(R)=R \ e^{it} \mid t \in [0,\pi][/itex] is a semicircular contour. Since we have [itex]f(z)=\frac{1}{z^4+1}[/itex], we can then use the Residue Theorem to evaluate the contour integral.

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