October 8, 2024
Introduction Hello everyone! In this blog post, we will discuss how to take the integral of $\sin 2x$. This is important to know, because it touches on two important concepts – integrals of trigonometric functions and $u$-substitution. Integrals of Trig Functions We know that the derivative of $\sin x$ is $\cos x$, and the derivative […]
September 10, 2024
Hello everyone! Today we will walk you through how to take the integral of $\csc(x)$. This is a highly unintuitive integral that cannot be solved by normal means, but $\csc(x)$ is also a very common function, so it’s important to know how to integrate it. The Integral First, we multiply the integrand of $\displaystyle\int\csc(x)\,dx$ by […]
September 10, 2024
Hello everyone! This article will walk you through how to take the integral of $x$. This is one of the most basic integrals, so it’s extremely important to know. Finding the Integral By the Fundamental Theorem of Calculus, $\displaystyle\dfrac{d}{dx} \int f(x)\,dx = f(x)$ for any continuous function $f(x)$. In other words, differention and integration are […]
September 4, 2024
You may remember from your math class that \(\displaystyle\int \cos x \, dx = \sin x + C\), but what happens if it is integral of -cos x? Negative sign is a multiplication by (-1) To get from $\cos x$ to $-\cos x$, we need to multiply it by a negative one: \[ \begin{align*} (-1) […]
September 4, 2024
To find the integral of x 2, written as \(\displaystyle \int x^2 \, dx\), we use the Power Rule for integration. The Power Rule for differentiation states: \[ \dfrac{d}{dx}(x^n) = n \cdot x^{n-1} \] To integrate, we reverse this process with what’s called the Reverse Power Rule: \[ \displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} […]
September 4, 2024
Have you ever wondered how to find the integral of \(\sqrt{x}\)? It’s actually quite simple once you know the power rule! Let’s start by rewriting \(\sqrt{x}\) in a way that’s easier to work with. We know that: \[ \sqrt{x} = x^{\frac{1}{2}} \] Now, we can use the power rule for integration. The power rule says […]
August 19, 2024
Because an integral is, in essence, the inverse function to a derivative, the integral of \(\cos x\) is going to be the function whose derivative is equal to \(\cos x\). To do this, let’s look at the derivatives for the basic trigonometric functions: \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin […]
April 9, 2024
In order to find the integral of 1 x 2, we will use the power rule. In other words, we want to use $$ \dfrac d{dx}x^n = nx^{n-1}$$ to find $$\displaystyle\int \dfrac{1}{x^2}dx$$ At first, it is not clear how the two are related. Recall, however, that another name for the integral is antiderivative. In other […]
April 9, 2024
In the realm of calculus, certain integrals stand out for their frequent application and the elegant techniques required for their resolution. One such integral is \( \int \dfrac{1}{x^2+1} dx \), notable for its direct connection to the arctangent function, a fundamental element in trigonometric calculations. The process of integrating \( \dfrac{1}{x^2+1} \) showcases the synergy […]
