# Tagged Questions

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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### How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
3answers
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### Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

I need to find a closed-form for the following integral. Please give me some ideas how to approach it: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
1answer
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### Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
0answers
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### Is there any closed form of an upper-bound of the following equation?

Could you please let me know if you can find the closed form of the following Equation (or any upper-bound that converges): $\sum_{i=1}^\infty(\dfrac{X}{Y^i})^i i!$, where $0<X<1$ and $Y>1$. ...
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### A strange integral having to do with the sophomore's dream:

I recently noticed that this really weird equation actually carries a closed form! $$\int_0^1 \left(\frac{x^x}{(1-x)^{1-x}}-\frac{(1-x)^{1-x}}{x^x}\right)\text{d}x=0$$ I honestly do not know how to ...
2answers
159 views