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 Marian – Mathematics

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 Marian – Mathematics

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 Marian – Mathematics

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 (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 Marian – Mathematics

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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 (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

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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
