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
\[\begin{align*} \displaystyle\boxed{\log_a b = x \text{ is equivalent to } a^x = b} \end{align*}\] What is $\log_a b$? This expression reads “log base a of b”. In mathematics, $\log_a b$ is called the logarithm of $b$ with base $a$. The logarithm $\log_a b$ answers the question: “What power do we need to raise $a$ to, in […]
October 28, 2024
$$\boxed{\log_b a = \dfrac{\log_k a}{\log_k b}}$$ The change of base formula for logarithms states that any $\log_b a$ can be expressed as $\dfrac{\log_k a}{\log_k b}$ where $k$ is any positive number. This formula is often used when calculating logarithms with the base that is inconvenient. What is a logarithm? A logarithm $\log_b a$ asks 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
$\log_b M + \log_b N = \log_b (M \times N)$ Let’s learn how to add logarithms. Don’t worry – it’s simple and straightforward. What is a Logarithm? A logarithm answers the question: To what exponent must we raise the base to get a certain number? For example: $\log_2 8 = 3$ because $2^3 = 8$. […]
October 25, 2024
The derivative of $\dfrac{x}{5}$ is $\boxed{\dfrac{1}{5}}$. To find the derivative of $\dfrac{x}{5}$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, $\dfrac{x}{5}$ can be rewritten as $\dfrac{1}{5} \cdot x^1$, where the coefficient $\dfrac15$ is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: […]
October 25, 2024
The derivative of $\dfrac{x}{3}$ is $\boxed{\dfrac{1}{3}}$. To find the derivative of $\dfrac{x}{3}$, we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n – 1} \end{align*}\] In this case, $\dfrac{x}{3}$ can be rewritten as $\dfrac{1}{3} \cdot x^1$, where the coefficient $\dfrac13$ is constant and \(x\) has an exponent of \(n=1\). Solution Using the power rule: […]
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 […]
