key term - ∫ ln x dx

Definition

The integral of the natural logarithm of x, or the indefinite integral of the natural logarithm function, is a fundamental operation in calculus that arises when evaluating integrals involving logarithmic functions. This term is particularly relevant in the context of integrals involving exponential and logarithmic functions, as covered in section 1.6 of the course material.

5 Must Know Facts For Your Next Test

1. The indefinite integral of the natural logarithm function, ∫ ln x dx, is a fundamental operation in calculus that arises when evaluating integrals involving logarithmic functions.
2. The antiderivative of the natural logarithm function, ln x, is $x \ln x - x + C$, where C is an arbitrary constant of integration.
3. The integral ∫ ln x dx can be used to find the area under the curve of the natural logarithm function, which has applications in various fields, such as economics and probability theory.
4. The integral ∫ ln x dx is closely related to the concept of the natural logarithm and its properties, such as the logarithm of a product being the sum of the logarithms of the factors.
5. The evaluation of ∫ ln x dx often involves the use of integration by parts, a technique that allows for the transformation of the integral into a more manageable form.

Review Questions

• Explain the relationship between the integral ∫ ln x dx and the natural logarithm function.
  • The integral ∫ ln x dx is directly related to the natural logarithm function, ln x. The natural logarithm function is the integrand, or the function being integrated, in the integral ∫ ln x dx. The antiderivative, or indefinite integral, of the natural logarithm function is $x \ln x - x + C$, where C is an arbitrary constant of integration. This relationship between the integral and the natural logarithm function is fundamental in evaluating integrals involving logarithmic functions.
• Describe how the integral ∫ ln x dx can be used to find the area under the curve of the natural logarithm function.
  • The integral ∫ ln x dx can be used to find the area under the curve of the natural logarithm function, ln x. This is because the indefinite integral represents the antiderivative of the natural logarithm function, which is $x \ln x - x + C$. By evaluating this antiderivative between two points, one can determine the area under the curve of the natural logarithm function over that interval. This application of the integral ∫ ln x dx has important uses in various fields, such as economics and probability theory, where the natural logarithm function and its area under the curve are relevant.
• Explain how the integration technique of integration by parts can be used to evaluate the integral ∫ ln x dx.
  • The integral ∫ ln x dx often requires the use of integration by parts, a technique that allows for the transformation of the integral into a more manageable form. In the case of ∫ ln x dx, one can let $u = \ln x$ and $dv = dx$, so that $du = \frac{1}{x} dx$ and $v = x$. Applying the integration by parts formula, the integral ∫ ln x dx can be evaluated as $x \ln x - \int \frac{x}{x} dx = x \ln x - x + C$, where C is the constant of integration. This demonstrates how integration by parts can be employed to solve the integral ∫ ln x dx, which is a crucial skill in evaluating integrals involving logarithmic functions.

Related terms

Indefinite Integral: An indefinite integral represents a family of antiderivatives, or functions whose derivative is the original function. It is denoted by the symbol ∫, followed by the integrand and dx.

Natural Logarithm:

The natural logarithm, denoted as ln x, is the logarithm with base e, the mathematical constant approximately equal to 2.71828.

Antiderivative:

An antiderivative, or primitive function, of a given function is a function whose derivative is the original function. The indefinite integral represents the set of all antiderivatives of a function.