key term - ∫ 1/x dx

Definition

The integral of the function 1/x, which represents the natural logarithm function. This integral is a fundamental operation in calculus and is particularly relevant in the context of integrals involving exponential and logarithmic functions.

5 Must Know Facts For Your Next Test

1. The integral of 1/x is the natural logarithm function, denoted as ln(x).
2. The natural logarithm function is the inverse of the exponential function with base e, meaning that e^(ln(x)) = x.
3. The integral of 1/x from a to b is equal to the difference between the natural logarithms of b and a, or ln(b) - ln(a).
4. The integral of 1/x dx represents the area under the curve of the function 1/x on the interval [a, b].
5. The natural logarithm function is a fundamental tool in calculus, with applications in areas such as growth and decay models, compound interest calculations, and the study of exponential and logarithmic functions.

Review Questions

• Explain the relationship between the integral of 1/x and the natural logarithm function.
  • The integral of the function 1/x, denoted as ∫ 1/x dx, is the natural logarithm function, ln(x). This is because the derivative of the natural logarithm function is 1/x, and the integral of 1/x is the antiderivative of this function, which is the natural logarithm. In other words, the integral of 1/x undoes the operation of taking the derivative of the natural logarithm function, making the natural logarithm the inverse function of the exponential function with base e.
• Describe the geometric interpretation of the integral of 1/x on the interval [a, b].
  • The integral of 1/x dx on the interval [a, b] represents the area under the curve of the function 1/x between the points a and b on the x-axis. This area is equal to the difference between the natural logarithms of b and a, or ln(b) - ln(a). This geometric interpretation is useful in understanding the properties and applications of the integral of 1/x, such as its use in calculating growth and decay rates, compound interest, and other exponential and logarithmic functions.
• Analyze the importance of the integral of 1/x in the context of integrals involving exponential and logarithmic functions.
  • The integral of 1/x, or ∫ 1/x dx, is a fundamental operation in calculus that is particularly relevant in the study of integrals involving exponential and logarithmic functions. This is because the natural logarithm function, which is the antiderivative of 1/x, is closely linked to the exponential function through their inverse relationship. Understanding the properties of the integral of 1/x, such as its connection to the natural logarithm and its geometric interpretation, is crucial for solving problems and analyzing the behavior of exponential and logarithmic functions in the context of integration.

Related terms

Natural Logarithm:

The logarithm function with base e, the mathematical constant approximately equal to 2.71828. The natural logarithm of a number x is denoted as ln(x).

Inverse Function: A function that undoes the operation of another function. The natural logarithm function is the inverse of the exponential function with base e.

Antiderivative:

A function whose derivative is the original function. The integral of 1/x is the antiderivative of the function 1/x, which is the natural logarithm function.