Integral of $\sin^2(x)$
The integral can be calculated using integration by parts (using the formula $∫u’(x)v(x)\,dx = u(x)v(x) - ∫ u(x)v’(x)\,dx$). Let’s (...)
July 3, 2017 – Jakub MarianMathematics
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 MarianMathematics
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 MarianMathematics
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 MarianMathematics
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, why the following equality (...)
June 29, 2017 – Jakub MarianMathematics

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Number of Nobel laureates per capita in Europe (map)
The following map shows the number of Nobel laureates per 10 million inhabitants in European countries (the smaller caption shows the (...)
August 29, 2016 – Jakub MarianMaps
The ‘day of the week boy or girl’ paradox explained
There is a well-known result in probability theory that seems to defy common sense. Note that in the problem statement we assume that the (...)
July 25, 2016 – Jakub MarianMathematics
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Labelling articles as “per capita” when the unit used is different
After I published my recent article about the Number of metal bands per capita in Europe, I got quite a few comments saying that my use of (...)
May 29, 2016 – Jakub MarianMathematics
The illusion of RGB screens
When an animal sees a computer screen, it doesn’t see the same colours as you do because our monitors only properly work for humans. This (...)
May 26, 2015 – Jakub MarianCognition, Physics
Difference between ‘violet’ and ‘purple’
People say that a picture is worth a thousand words, so let’s take a look at the two colours in comparison (there are various shades of (...)
May 17, 2015 – Jakub MarianCognition, Physics