November 8, 2024

Finding the Integral of 2x ln(x)

\[\begin{align*} \boxed{\int 2x \ln(x) \, dx = x^2 \ln(x) – \frac{x^2}{2} + C} \end{align*}\] 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…

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Finding the Derivative of 2x ln(x)

\[\begin{align*} \boxed{\frac{d}{dx} \left( 2x \ln(x) \right) = 2 \ln(x) + 2} \end{align*}\] Introduction To find the derivative of \( 2x \ln(x) \), we will use the product rule. Learning to work with logarithmic expressions is important in calculus as they show up later when tackling integration. Step-by-Step Solution The product rule, which states that if…

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How to Solve 6^3

\[\begin{align*} \boxed{6^3 = 6 \times 6 \times 6 = 216} \end{align*}\] Introduction The expression \( 6^3 \) is an example of exponential notation, where: – \( 6 \) is the base, which is the number being multiplied. – \( 3 \) is the exponent, which tells us how many times to multiply the base by…

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How to Solve 10^5

\[\begin{align*} \boxed{10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000} \end{align*}\] Introduction Exponents provide a compact way to express large numbers through repeated multiplication. The expression \( 10^5 \) is an example of exponential notation, where: – \( 10 \) is the base, representing the number being multiplied. – \( 5…

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Integration Rules

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…

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Understanding the Definition of an Integral

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…

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How to Solve 2^3

\[\begin{align*} \boxed{2^3 = 2 \times 2 \times 2 = 8} \end{align*}\] Introduction Exponents are a convenient way to represent repeated multiplication, and understanding how to evaluate them is essential in math. The expression \( 2^3 \) is an example of exponential notation, where \( 2 \) is the base, which is the number being multiplied,…

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Logarithm Rules

Introduction Logarithms are the inverse operations of exponents, allowing us to “undo” exponentiation and solve equations involving exponential terms. We will cover the main rules of logarithms, which make it easier to manipulate logarithmic expressions. Definition of Logarithms A logarithm answers the question: “To what exponent must a base be raised to get a certain…

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Understanding the Definition of Exponents

Introduction Exponentiation is a notation in mathematics representing repeated multiplication. They provide a way to express large numbers more concisely and simplify calculations in algebra, calculus, and many applied fields. Definition of Exponents An exponent is a small number placed above and to the right of a base number, indicating how many times to multiply…

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Exponent Laws

Exponent laws/rules are useful tools when evaluating expressions. They include the product rule, quotient rule, and power rules. Given integers $m$ and $n$, we have the following: Product Rule \[\begin{align*} x^m \cdot x^n = x^{m + n} \end{align*}\] This works as if we expand the exponent out as multiplications of $x$, then $x^m = (x…

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