Integral of $\sin^2(x)$
The integral can be calculated using integration by parts (remember that $∫u’(x)v(x)\,dx = u(x)v(x)  ∫ u(x)v’(x)\,dx$):
$$ ∫ (...)
July 3, 2017 – Jakub Marian – Mathematics

Integral of $\tan(x)$
The easiest way to integrate $\tan(x)$ is to recall that $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Do you see the necessary substitution? By choosing $y = \cos(x)$, that is, $dy = \sin(x)\,dx$ (...)
July 2, 2017 – Jakub Marian – Mathematics

Derivative of $x^x$
Calculating the derivative of $x^x$ is a very simple task, but it may be hard to find the general idea on your own, so here it is. We will (...)
July 2, 2017 – Jakub Marian – Mathematics

Integral of $x^n$
How to calculate the indefinite integral of $x^n$? It’s quite simple. We are looking for a function $f(x)$ such that $f’(x) = x^n$. As you (...)
July 2, 2017 – Jakub Marian – Mathematics

Integral of $e^{x^2}$ from $\infty$ to $\infty$
In this article, I will give a detailed explanation of why the Gaussian integral is equal to $√\pi$, that is, of the following famous (...)
June 29, 2017 – Jakub Marian – Mathematics

Correlatives in Esperanto
In the grammar of Esperanto, words like “where”, “there”, “somewhere”, “when”, “then”, “sometime”, etc., are called correlatives. (...)
June 27, 2017 – Jakub Marian – Esperanto

‘June’ in European languages
The English name “June” comes from Latin iunius, the month of the Roman goddess Iuno (usually spelled “Juno” in English), who was the (...)
June 7, 2017 – Jakub Marian – Maps

Gold reserves of national central banks per capita in Europe
Most national central banks in Europe hold a large amount of reserves in the form of foreign banknotes and deposits, treasury bills, gold, (...)
May 28, 2017 – Jakub Marian – Maps

Names of European languages in those languages
The proper term used in linguistics for the name of a language in that very same language is “autonym” (or “endonym”), so the (...)
May 11, 2017 – Jakub Marian – Maps

‘May’ in European languages
The month of May is named after Maia, the Roman goddess of growth (her name was thought to be related to the Latin word maior, “larger, (...)
May 4, 2017 – Jakub Marian – Maps
