For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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6
votes
0answers
243 views
+300

Integration of product of functions(Special form)

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
19
votes
2answers
332 views
+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
13
votes
3answers
507 views
+100

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
15
votes
1answer
201 views
+300

Closed form for ${\large\int}_0^1\frac{\ln^{\color{magenta}3}x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
10
votes
0answers
96 views
+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
5
votes
1answer
54 views
+300

Simplification of an expression containing $\operatorname{Li}_3(x)$ terms

In my computations I ended up with this result: $$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102 ...
2
votes
0answers
40 views
+50

On a property of polylogarithm

I have an observation, and I don't know that the following statement is true or not. If not give a counterexample, if it is true prove it, or give a reference about it. Let $n \in \mathbb{R}$, $z \in ...