Problem 34 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{d x}{x \ln x^{2}}, x>1 $$

Step-by-Step Solution

Answer

The evaluated integral is \( \frac{1}{2} \ln (\ln x) + C \).

Step 1: Identify the Integral

The given integral is \( \int \frac{d x}{x \ln x^{2}} \). We need to find an antiderivative for this function.

Step 2: Simplify the Expression

Notice that \( \ln x^{2} \) can be rewritten using the logarithm power rule as \( 2 \ln x \). Hence, the integral becomes \( \int \frac{dx}{2x \ln x} \).

Step 3: Choose a Substitution

Let \( u = \ln x \). Then, the differential \( du = \frac{dx}{x} \). The integral becomes \( \int \frac{du}{2u} \).

Step 4: Integrate with Substitution

The integral \( \int \frac{1}{2u} du \) is \( \frac{1}{2} \ln |u| + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back the Variable

Replace \( u \) with \( \ln x \): \( \frac{1}{2} \ln |\ln x| + C \). Since \( x > 1 \), \( \ln x \) is positive, and we can simplify this to \( \frac{1}{2} \ln (\ln x) + C \).

Step 6: Verify by Differentiation

Differentiate the result \( \frac{1}{2} \ln (\ln x) + C \) with respect to \( x \). The derivative is \( \frac{1}{2} \cdot \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{2x \ln x} \), which matches the integrand when multiplied by 2, confirming the correctness of the antiderivative.

Key Concepts

Antiderivative

In calculus, finding an antiderivative is an essential skill. An antiderivative, also known as an indefinite integral, is a function whose derivative gives back the original function. In mathematical notation, if we have a function \( F(x) \), its derivative \( F'(x) \) becomes the original function \( f(x) \). Therefore, to find an antiderivative of a function, you need to determine another function that differentiates back to it.
By determining the antiderivative, you can solve integrals and make calculative predictions about quantities like area under curves. In the exercise, the function \( \int \frac{dx}{x \ln x^{2}} \) simplifies to finding the antiderivative of \( \frac{1}{2u} \) after substitution. This leads to \( \frac{1}{2} \ln |u| + C \), where \( C \) is an integration constant.

Substitution Method

The substitution method is a powerful technique in integral calculus used to simplify complex integrals. It involves changing the variables to transform a difficult integral into an easier one, often making the process of integration more straightforward.
Steps for Substitution:

• Identify a part of the integral that can be replaced by a single variable, such as \( u \).
• Express the differential, such as \( du \), in terms of \( dx \).
• Substitute these expressions into the integral to simplify it.
• After integrating, replace the substitution variable \( u \) with the original expression.

In our given example, the substitution \( u = \ln x \) made it easier to handle the logarithmic function, transforming the integral to \( \int \frac{du}{2u} \), which is simpler to integrate.

Integral Calculus

Integral calculus is a branch of calculus focusing on integrals and their properties. It deals with the process of finding antiderivatives, as well as determining areas, volumes, and accumulations.
There are two types of integrals:

• Indefinite Integrals: These find a family of functions, including an arbitrary constant \( C \), and represent the general form of antiderivatives.
• Definite Integrals: These evaluate the area under a curve between specified bounds, providing a numerical result without an arbitrary constant.

By determining the integral of \( \int \frac{dx}{x \ln x^{2}} \), both concepts of indefinite integration and simplifying with calculus techniques such as substitution come together. The resulting antiderivative points to the broader usefulness of integral calculus in solving real-world problems by providing general solutions and specific value estimates.

Logarithmic Functions

Logarithmic functions, represented as \( \ln(x) \) for natural logarithms, are intrinsic to calculus because of their unique properties and derivatives. They arise often in integrals due to their simplicity for differentiation.

Key characteristics of logarithmic functions include:

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• The logarithmic power rule allows simplifying expressions such as \( \ln(x^2) = 2\ln(x) \).
• They grow slowly compared to polynomials or exponential functions.

In the provided exercise, recognizing that \( \ln(x^2) = 2\ln(x) \) was crucial to simplifying the integral. This simplification enabled the use of substitution, smoothing the path uniquely afforded by the properties of logarithmic functions.