Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int x \ln 2 x d x $$

Step-by-Step Solution

Answer

\(u = \ln(2x)\) and \(d v = x d x\).

Step 1: Identify \(u\) and \(d v\)

The integrand is \(x \ln(2x)\). From the function, you can recognize \(x\) and \(\ln(2x)\). It is known that the derivative of \(\ln(2x)\) is simpler than the function itself but for \(x\) it will only be a constant when differentiated. Thus, \(u = \ln(2x)\) and \(d v = x d x\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

\(u = \ln(2x)\) and \(d v = x d x\).

Key Concepts

Integral Calculus

Integral Calculus is a core area of calculus that focuses on finding functions that can describe the accumulation of quantities and how they change. A key concept in integral calculus is the integral, which can be understood as the reverse operation to differentiation. An integral sums up or "accumulates" values across a range.

There are two primary types of integrals: definite and indefinite. The indefinite integral represents a function family, providing antiderivatives of a given function. In contrast, the definite integral calculates the "area" under the curve between two specific points, giving a numeric result.

Integral calculus is widely used in physics, engineering, and economics to solve problems involving areas, volumes, displacement, and much more. Tools such as the Fundamental Theorem of Calculus establish connections between differentiation and integration, making computations more efficient.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they play an essential role in calculus and algebra. The logarithm of a number is the power to which the base must be raised to obtain that number.

In the expression \(\ln(2x)\), the natural log function, or \(\ln\), is used, which is logarithm with base e, where e approximately equals 2.718. Logarithmic functions, especially natural logs, have special properties that simplify the processes of differentiation and integration.

Understanding how to manipulate logarithmic expressions is key to solving problems involving integration and differentiation, as these functions often simplify more complex expressions. Common properties, such as \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b\ln(a)\), help in breaking down complex problems into more manageable parts.

Differentiation

Differentiation is the process of finding the derivative of a function, representing how a function changes. It measures the rate of change or the slope of the function's graph.

In the problem that you are solving with integration by parts, you need to differentiate the function \(\ln(2x)\). Differentiating \(\ln(2x)\) provides insight into how integration by parts will proceed and confirm why it is chosen as \(u\) in the process.

The derivative \(\frac{d}{dx} [\ln(2x)] = \frac{1}{2x} \cdot 2 = \frac{1}{x}\) shows how applying basic differentiation rules can simplify calculations. In integration by parts, this derivative will play a crucial role while evaluating the integral to determine the remaining parts of the function.

Definite Integrals

Definite integrals are used to calculate the total accumulation of quantities where limits of integration are specified, providing the area under a curve between two points. Unlike indefinite integrals, which yield a general formula for antiderivatives, definite ones result in a specific number.

In practical applications, definite integrals have broad usage, such as finding areas, volumes, work done by forces, or total growth over time. Their calculation often utilizes boundaries known as the limits of integration, setting the scope of accumulation in quantitative analysis.

The formal representation is \(\int_a^b f(x) \ dx\), where \(a\) and \(b\) are the bounds, showing that integration doesn't just solve for a function but also provides specific quantifiable results. Integration by parts can also be used to solve definite integrals effectively, transforming challenging equations into easier solvable forms by cleverly choosing parts of the integrand to differentiate and integrate.