December 11, 2024
Integration by parts is the inverse of the derivative product rule. It is very useful when $u$-substitution and using standard integration techniques aren’t enough to handle the problem. Integration by parts states that if you have an integral of the form $\displaystyle \int u \: dv$, we may rewrite it as: \[\begin{align*} \int u \: […]
December 11, 2024
There are many different ways to compute integrals. While all of the integration rules are important, they all have their place and can be used in different situations. When it comes to integrating polynomials, the inverse power rule is the most useful technique. The inverse power rule states: \[\begin{align*} \boxed{\int x^n \: dx = \dfrac{x^{n+1}}{n+1} […]
December 5, 2024
\[\begin{align*} \int \sec^2(x) dx = \boxed{\tan(x) + C}. \end{align*}\] To find the indefinite integral of $\sec^2(x)$, let’s first recall that the derivative of $\tan x$ is \[\begin{align*} \dfrac{d}{dx}(\tan(x)) = \sec^2(x) \end{align*}\] The First Fundamental Theorem of Calculus states that \[\begin{align*} f(x) = \dfrac{d}{dx} \displaystyle\int_a^x f(x) dx. \end{align*}\] In other words, derivatives and integrals are inverse […]
November 18, 2024
\[\begin{align*} \boxed{\int \sec(2x) \: dx = \frac{1}{2}\ln \left|\tan \left(2x\right)+\sec \left(2x\right)\right|+C} \end{align*}\] where \( C \) is the constant of integration. Solving for the Integral To solve many integrals that involve composite functions, we turn to $u$-substitution. It is a little easier to work with $\sec(x)$ instead so we set $u = 2x$ and $du = […]
November 8, 2024
\[\begin{align*} \boxed{\int 2x \ln(x) \, dx = x^2 \ln(x) – \frac{x^2}{2} + C} \end{align*}\] where \( C \) is the constant of integration. Introduction In calculus, integrating functions that combine both polynomial and logarithmic terms, like \( 2x \ln(x) \), requires the specific technique of integration by parts. This method is particularly useful for integrating […]
November 8, 2024
Introduction Integration is one of the two main operations in calculus, the other being differentiation. Integrals allow us to calculate areas, volumes, accumulated quantities, and more. To simplify integration, mathematicians have developed a set of integration rules. These rules make it easier to work with different types of functions and solve integrals efficiently. In this […]
November 8, 2024
Introduction The concept of an integral is a fundamental part of calculus and mathematical analysis, allowing us to measure quantities that accumulate over time or space. Integrals are used extensively in physics, engineering, economics, and many other fields to calculate areas, volumes, and total values of quantities. In this article, we’ll cover the basic definition […]
November 7, 2024
The integral \( \displaystyle \int \sin x \, dx \) is a basic result in calculus, and is used in various calculations involving trigonometry, wave motion, and oscillatory behavior. Solution The antiderivative of \( \sin x \) with respect to \( x \) is: \[\begin{align*} \displaystyle \int \sin x \, dx = -\cos x + […]
November 7, 2024
The integral \( \displaystyle \int \dfrac{\ln x}{x} \, dx \) is an example of integrating compund functions, a foundational skill in calculus. Further, $\ln x$ is a function that arises in various real-world contexts, as well as in classroom and test settings. Solution The antiderivative of \( \dfrac{\ln x}{x} \) with respect to \( x […]
