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1d
revised Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
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1d
revised Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
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1d
answered Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Dec
17
comment How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?
@FreeMind I don't know of a list, but basically any related series can be derived by differentiating or integrating the series in my post or the series $$\sum_{k=0}^{\infty} b^{k} \sin(k \theta) = \frac{b \sin \theta}{1-2b \cos \theta + b^{2}} \ , \ |b| <1$$ with respect to $\theta$. For example, integrating the above series with respect to $\theta$ we get $$\sum_{k=1}^{\infty} \frac{b^{k} \cos (k \theta)}{k} = - \frac{1}{2} \log \left(1- 2 b \cos \theta + b^{2} \right) \ , \ |b| <1.$$
Dec
16
comment Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$
@Venus Thanks. I had accidentally put the parameter $a$ in the numerator and the parameter $b$ in the denominator.
Dec
16
revised Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$
I switched the parameters
Dec
16
comment How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?
@FreeMind $$ \begin{align} \sum_{k=0}^{\infty} b^{k} \cos(k \theta) &= \text{Re} \sum_{k=0}^{\infty} b^{k} e^{ik \theta} = \text{Re} \sum_{k=0}^{\infty} (be^{i \theta})^{k} \\ &= \text{Re} \ \frac{1}{1-be^{i \theta}} \\ &= \text{Re} \ \frac{1-be^{-i \theta}}{(1-be^{i \theta})(1-be^{-i \theta})} \\ &= \text{Re} \ \frac{1- b \cos \theta + i b \sin \theta}{1 - 2 b \cos \theta + b^{2}} \\ &=\frac{1- b \cos \theta}{1-2b \cos \theta + b^{2}} \end{align}$$
Dec
16
answered Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$
Dec
13
comment About $\int_0^{\pi/2}\arctan(1-\sin^2 (x) \cos^2 (x))dx = \pi \left( \frac{\pi}{4}-\pi \arctan \sqrt{\frac{\sqrt{2}-1}{2}}\right)$
Very clever use of the sum identity for arctan. +1
Dec
12
comment A couple of definite integrals related to Stieltjes constants
@IaroslavBlagouchine Thank you.
Dec
12
revised A couple of definite integrals related to Stieltjes constants
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Dec
12
revised A couple of definite integrals related to Stieltjes constants
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Dec
12
answered A couple of definite integrals related to Stieltjes constants
Dec
12
answered How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?
Dec
10
revised A definite integral with hyperbolic cosines
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Dec
10
revised Evaluating $\int_{-\infty}^{\infty} \frac{\log^{2}(1+ix^{2})}{1+ix^{2}} dx $ using contour integration
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Dec
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awarded  Caucus