October 29, 2024
Results Summary If you’re searching for “integrate 2 1 x,” here are some possible solutions: $$\boxed{\int \frac{2}{1 + x} \, dx = 2 \ln |1 + x| + C}$$ $$\boxed{\int \frac{2}{1 – x} \, dx = -2 \ln |1 – x| + C}$$ $$\boxed{\int \left( 2 + \frac{1}{x} \right) \, dx = 2x + \ln […]
October 29, 2024
Results Summary If you’re searching for “integral of 2 x,” here are some possible solutions: $$\boxed{\int 2x \, dx = x^2 + C}$$ $$\boxed{\int \frac{2}{x} \, dx = 2 \ln |x| + C}$$ $$\boxed{\int (2 + x) \, dx = 2x + \frac{x^2}{2} + C}$$ Below are three potential integrals based on common mathematical usage […]
October 29, 2024
Results Summary If you’re searching for “integral of 1 1 x 2,” here are some possible interpretations and their solutions: $$\boxed{\int \frac{1}{1 + x^2} \, dx = \arctan(x) + C}$$ $$\boxed{\int \frac{1}{1 – x^2} \, dx = \frac{1}{2} \ln \left| \frac{1 + x}{1 – x} \right| + C}$$ Let’s go through each interpretation and solve […]
October 28, 2024
Introduction Before diving into calculus and memorizing formulas, it is important to take a step back and understand what an integral actually means. What is an Integral? An integral, graphically, represents the area under a curve. You might be wondering why it’s useful to know the area under the curve. The reason is that the […]
October 25, 2024
If $f$ is continuous on $[a, b]$ then there exists a number $c$ in $(a, b)$ such that: \[ \int_a^b f(x) \, dx = f(c)(b – a) \] Understanding the Mean Value Theorem for Integrals Hello students! Today, we will learn about the Mean Value Theorem for Integrals. This theorem helps us find the average […]
October 25, 2024
The integral of $\dfrac{x}5$ is $\boxed{\dfrac{x^2}{10} + C}$. To find $\displaystyle \int \dfrac{x}5 \: dx$, we may pull out the constant $\dfrac15$. After doing so, we end up having to find the integral of $x$ using the power rule: \[\begin{align*} \int x^n \: dx &= \dfrac{x^{n + 1}}{n + 1} + C \end{align*}\] Plugging in […]
October 25, 2024
The integral of $\dfrac{x}3$ is $\boxed{\dfrac{x^2}{6} + C}$. To find $\displaystyle \int \dfrac{x}3 \: dx$, we may pull out the constant $\dfrac13$. After doing so, we end up having to find the integral of $x$ using the power rule: \[\begin{align*} \int x^n \: dx &= \dfrac{x^{n + 1}}{n + 1} + C \end{align*}\] Plugging in […]
October 8, 2024
Fundamental Theorem of Calculus: Part one: $$\int_a^bf(x)\:dx=F(b)-F(a)$$ Where $F(x)$ is an antiderivative of $f(x)$ Part two: $$\dfrac{d}{dx}\int^x_af(t)\:dt=f(x)$$ Basic Integration Rules: Constant Rule: $\displaystyle{\int} 0 \:dx =C$ Constant Multiple Rule: $\displaystyle{\int} Cf(x) \:dx = C\displaystyle{\int}f(x) \:dx$ Sum and Difference Rule: $\displaystyle{\int}\left[ f(x)\pm g(x) \right]\:dx = \displaystyle{\int}f(x) \:dx\pm \displaystyle{\int}g(x)\:dx$ Power Rule: $\displaystyle{\int}x^n \:dx= \dfrac{x^{n+1}}{n+1}+C$ Inverse Bounds Rule: […]
October 8, 2024
Hello everyone! In this article, we will go over how to take the integral of $\dfrac{x-1}{x+1}$. The technique shown in this article can also be used for integrating many fractions of rational equations, so it’s definitely a handy tool to know. Taking the Integral of x-1 over x+1 We will start expressing $\dfrac{x-1}{x+1}$ in a […]
