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11h
awarded  Nice Answer
11h
answered Integrating linear/trigonometric
Aug
24
answered Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$
Aug
24
comment Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$
Are you at all familiar with the Nielsen generalized polylogarithm? I'm currently attempting a response that makes heavy use of them, and I'm trying to decide how much background I should include on these functions. The chief advantage of these functions here is MUCH tidier anti-derivative than the unholy mess you mentioned above.
Aug
23
comment Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$
You have a bad habit of writing terribly interesting questions about polylogarithms without tagging them as such, with the result that I can never find them again in the vast ocean of "integration" tags. I have decided it's high time I fixed that. ;)
Aug
23
revised Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$
Added polylog tag
Aug
17
comment Find $\int_0^\frac{\pi}{2} \frac {\theta \cos \theta } { \sin \theta + \sin ^ 3 \theta }\:d\theta$
+1) I think this answer can also serve as a sort of life-lesson for the high school student: if your calculus teacher is having trouble solving an integral, try differentating under the integral sign. Why does this have such a good chance of working? Because for some some inscrutable reason, THEY DON'T censored TEACH IT! and it's likely the one trick they haven't tried. :<
Aug
16
comment $\int_{0}^{\infty}\frac{\ln x dx}{x^2+2x+2}$
My favorite :) +1
Aug
11
comment A countor integral involving a branch cut
@Leucippus Are you committed to a contour integral approach? If you're main goal is to evaluate the integral, there's a very nice way to solve it via elementary methods.
Aug
11
comment Sum of an infinite series.
FYI, this kind of series is called an arithmetico-geometric series. This has to be one of the most do everything-to-death math probelms on the Internet, and I suspect the main reason students have so much trouble looking up this problem for themselves is that they simply don't know what to call it. There is a nice Wikipedia page devoted to the topic: en.m.wikipedia.org/wiki/Arithmetico-geometric_sequence
Aug
11
answered Sum of Legendre function
Aug
11
comment Finding $\int \sec^3x\,dx$
see also the Wikipedia page en.m.wikipedia.org/wiki/Integral_of_the_secant_function.
Aug
6
awarded  Necromancer
Aug
6
awarded  Revival
Aug
6
answered Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$
Aug
5
comment Closed form for an infinite product
@wltrup As it so happens, Ramanujan is one of the few mathematicians I know of who literally would prove identities in his sleep on the regular. He claimed the goddess Namigiri would reveal them to him in his dreams, also the only case of divine revelation of non-trivial mathematical results I've ever been able to find. TMYK :)
Aug
4
awarded  Nice Answer
Aug
3
comment Closed form of series involving Fibonacci numbers
Very nicely done! Your strategy is particularly appealing to me given my affection for logarithmic integrals. Any chance an integral of that type has already been solved here?
Jul
31
comment Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
@user153012 I retract my previous statement. I had the same starting point and plan of attacking as Vlad, but he implemented it with far more dexterity than I would have been able to. :)
Jul
31
comment Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$
@Lucian I've long suspected that to be one of the central algorithms employed by Mathematica ;)