Science

 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 (using the formula $∫u’(x)v(x)\,dx = u(x)v(x) - ∫ u(x)v’(x)\,dx$). Let’s (...) 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, why the following equality (...) June 29, 2017 – Jakub Marian – Mathematics