How to calculate the factorial of 1/2
As I proved in my article on integration by parts (at the end), the factorial of a natural number $n$ can be calculated using the (...)
July 8, 2017 – Jakub Marian – Mathematics

Integral of $\log(x)$ (or $\ln(x)$)
The natural logarithm (of base $e$) is usually denoted $\log(x)$ in mathematics, but for the sake of students of other disciplines, we (...)
July 7, 2017 – Jakub Marian – Mathematics

Explanation of integration by parts
Integration by parts is one of the first methods people learn in calculus courses. Together with integration by substitution, it will (...)
July 6, 2017 – Jakub Marian – Mathematics

Integral of $\sin^3(x)$
Integrating the third power of $\sin(x)$ (or any odd power, for that matter), is an easy task (unlike $∫ \sin^2(x)\,dx$, which requires (...)
July 4, 2017 – Jakub Marian – Mathematics

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

‘Merry Christmas’ in European languages (map)
A map showing how to wish people a “Merry Christmas” in French, German, Spanish, Italian and other European languages. (...)
December 23, 2016 – Jakub Marian – English, French, German, Italian, Maps, Other languages, Russian, Spanish
