Although we are not so numerous as other respected professionals, like for example lawyers, I wonder if we could come up with a reasonable estimate of our population. Needless to say, the question more or less amounts to the definition of"mathematician". Since I should like to count only resea...

My objective: Use cylindrical coordinates to find the volume of the prism whose base is the triangle in the xy-plane given by y = 0, x = 1, and y = $\frac{7}{2} x$, and whose top is given by z = 8 - y. In Cartesian coordinates, this is straightforward to me: $\int_{0}^{1}\int_{0}^{7/2x}\in...

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ dx$$ As regards the first integral, my approach was to combine the integration by parts and the va...

This is an integral coming from personal research, and very important to me, but it does not seem an easy job to do. If a solution is not possible then I'd be glad with a closed form only. $$\int_{[0,1]^2} \frac{(1-x-y+x y+x \log(x)-x y\log(x)+y \log(y)- x y\log(y)+x y\log(x)\log(y))\log(1+x y)}...

I sometimes find writing and keeping track of the constants of integration a somewhat messy job. Yes, sometimes it's necessary but in many situations that I come across in my level of mathematics, it is a waste of time and space. An exagerated example: $$f''(x) = g''(x)\\ \stackrel{\int\text{in...