November 7, 2024

Integral of sin x Explained

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 +…

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Integral of \( \dfrac{\ln x}{x} \): How to Solve It

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…

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Antiderivative of \( \dfrac{1}{x} \): A Quick Guide

The antiderivative of \( \dfrac{1}{x} \) is a foundational result in calculus, and appears in a variety of contexts. Specifically, it reveals interesting properties of the natural logarithm in calculus. Solution The antiderivative of \( \dfrac{1}{x} \) with respect to \( x \) is: \[\begin{align*} \displaystyle \int \dfrac{1}{x} \, dx = \ln |x| + C…

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Finding the Derivative of \(\ln x\)

Answer \[\begin{align*} \boxed{\frac{d}{dx}(\ln x) = \frac{1}{x}} \end{align*}\] The Derivative of \(\ln x\) The function $\ln x$ is called the natural logarithm function. More specifically, $\ln x = \ln_e x$, where $e \approx 2.718$. The derivative of $\ln x$ is: \[\begin{align*} \frac{d}{dx}(\ln x) = \frac{1}{x} \end{align*}\] However, some sources also use $\log x$ to describe $\ln…

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Derivative of Exponential Functions

Introduction In calculus, an exponential function refers to any function that is a base with an exponent of some expression of $x$. Examples include $f(x) = 2^x$, $3^x$, $e^{x – 1}$, $5^{\sqrt{x + 1}}$, and so on. Derivatives of exponential functions often appear in calculus. The Key Idea The derivative of an exponential function $f(x)…

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Derivative of exp

Introduction Derivative of “exp” means a derivative of $e^x$, where the number $e$ is an Euler’s constant, roughly equal to $2.71828$. As an equation, \[\begin{align*} \exp(x) = e^x, \text{where e} \approx 2.71828. \end{align*}\] Derivative of $e^x$ The derivative of the function is equal to the slope of the function at a given point. The derivative…

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$e$ to the $\ln x$

\[\begin{align*} \boxed{ e^{\ln x} = x } \end{align*}\] What is $e$? \( e \) is a special number in math, approximately equal to 2.718. It is also called Euler’s constant and is the base of a logarithm called the natural logarithm. You can think of it as the number that makes many things in math…

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$\log_b(b^k)$ Rule

\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\] This rule says that if we take the log of a number that is to the $k$th power of the base $b$, the result is simply the exponent $k$. In other words: \[\begin{align*} \log_{b}(b^k) = k \end{align*}\] This rule works because the logarithm $\log_b(x)$ asks, “What power must we raise…

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Evaluating $\log_{10}(1000)$

What does $\log_{10}(1000) = X$ mean? We start with the expression: \[\begin{align*} \log_{10}(1000) = X \end{align*}\] This equation means that we are looking for a value of \(X\) such that: \[\begin{align*} 10^X = 1000 \end{align*}\] In other words, \(\log_{10}(1000)\) represents the power to which we must raise 10 to obtain 1000. Our goal is to…

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