Problem 68 - Applied Calculus

Question

Which one of these formulas is correct? a. \(\int \ln x d x=\frac{1}{|x|}+C\) b. \(\int \ln |x| d x=\frac{1}{x}+C\) c. \(\int \frac{1}{x} d x=\ln |x|+C\) d. \(\int \frac{1}{\ln x} d x=|x|+C\)

Step-by-Step Solution

Answer

Option c is correct: \(\int \frac{1}{x} dx = \ln |x| + C\).

Step 1: Understand the Integral of \( rac{1}{x} \)

The integral \( \int \frac{1}{x} \, dx \) has the antiderivative \( \ln |x| + C \), where \( C \) is the constant of integration. This is a well-known result in calculus.

Step 2: Review Option C

Option c is given as \( \int \frac{1}{x} \, dx = \ln |x| + C \). This matches the standard result from the previous step.

Step 3: Analyze Other Options

Options a, b, and d propose incorrect antiderivatives, as they do not match known results for their respective integrals. For example, \( \int \ln x \, dx \) and \( \int \ln |x| \, dx \) do not yield the proposed results in a or b.

Step 4: Confirm the Solution

Based on Steps 1 and 2, option c is confirmed as correct. Option c states the standard solution for the integral of \( \frac{1}{x} \).

Key Concepts

Antiderivative

The concept of an antiderivative is a fundamental aspect of calculus. Simply put, an antiderivative of a function is another function whose derivative returns the original function. When we talk about finding an antiderivative, we are essentially performing an operation that is the reverse of differentiation. Given a function \( f(x) \), an antiderivative is a function \( F(x) \) such that its derivative \( F'(x) \) is \( f(x) \). For instance, when we say \( \int \frac{1}{x} \, dx = \ln |x| + C \), we identify that the derivative of \( \ln |x| \) is \( \frac{1}{x} \).

The process of finding antiderivatives is also known as integrating, and this is where integral calculus comes into play. Calculus students often encounter the challenge of discovering various antiderivatives across different types of functions. Understanding the rules for differentiation helps in identifying and verifying these antiderivatives. Remember, every differentiable function has infinitely many antiderivatives because of the arbitrary constant we introduce, which is known as the constant of integration.

Integral Calculus

Integral calculus is a branch of mathematics focused on integrals and their properties. It is an essential part of calculus along with its counterpart, differential calculus, which deals with derivatives. In simple terms, while differential calculus helps us find how things change, integral calculus helps determine how much there actually is, for example, in terms of area or volume.

One of the primary tools in integral calculus is the definite integral, which calculates the exact area under a curve, often between two points on the x-axis. Another major component is the indefinite integral, which comes into play when determining antiderivatives. This is where the integral symbol \( \int \) appears in conjunction with a function and a differential, such as \( dx \), representing the variable of integration.

Integral calculus also finds applications outside computing areas; it can be used to solve problems related to physics, engineering, statistics, and various other fields by providing a mathematical approach to real-world scenarios.

Constant of Integration

The constant of integration plays a crucial role in the calculus landscape, especially when dealing with indefinite integrals. Whenever an antiderivative is found through integration, it consists of a general form plus this constant, which we denote as \( C \).

The reason this constant exists is that an infinite number of antiderivative functions can satisfy the derivative condition for a given function. Imagine drawing a curve and shifting it up or down vertically; any of those shifted curves still represent valid solutions for the antiderivative of the original function. By adding a constant \( C \), mathematicians account for all these possible vertical shifts.

• It ensures completeness in the solution.
• Represents infinitely many solutions.
• Is important when solving differential equations as initial conditions help determine its value.

The constant of integration essentially serves as a reminder of the richness of possible solutions an antiderivative can have.