Mathematics

What is $0^0$ ($0$ to the power $0$)?
A short answer to this question is that $0^0$ is usually left undefined in introductory calculus classes, just like $0/0$, but it is often (...)
October 15, 2017 – Jakub MarianMathematics
Integral of $\cot(x)$
To integrate cot(x), recall that cot(x) = cos(x)/sin(x), so ∫ cot(x) dx = ∫ cos(x)/sin(x) dx. (...)
September 21, 2017 – Jakub MarianMathematics
Integral of $\sin(x)\cos(x)$
The function $\sin(x)\cos(x)$ is one of the easiest functions to integrate. All you need to do is to use a simple substitution $u = (...)
September 6, 2017 – Jakub MarianMathematics
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 MarianMathematics
Integral of $\log(x)$ (or $\ln(x)$)
The natural (base $e$) logarithm is usually denoted $\log(x)$ in mathematics, but for the sake of students of other disciplines, we will (...)
July 7, 2017 – Jakub MarianMathematics
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 MarianMathematics
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 MarianMathematics
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 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