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The substitution method is a common technique in calculus used to simplify integrals by changing variables. This method involves substituting a part of the integral with a new variable, often denoted as \( u \). This can transform a complex integral into a simpler one, making it easier to evaluate.

Here is how to use the substitution method:

• Identify a substitution. Look for a function and its derivative within the integral that can simplify the expression when replaced with \( u \).
• Differentiate the substitution. If you choose \( u = f(x) \), find \( du = f'(x) \, dx \). This helps in writing the integral in terms of \( u \).
• Replace and Simplify. Substitute \( u \) and \( du \) into the integral, simplifying it.
• Integrate. Solve the simpler integral in terms of \( u \).
• Substitute back. Replace \( u \) with the original expression in terms of \( x \) to find the final solution.

In our exercise, \( u = \ln x \) was chosen because its derivative \( \frac{1}{x} \, dx \) appeared in the integral. This made the substitution a perfect fit, turning the original integral into \( \int u^2 \, du \), which is much simpler to solve.

A definite integral computes the area under a curve from one specific point to another along the x-axis. It is represented by the integral symbol with upper and lower bounds (limits).

Characteristics of definite integrals include:

• Has limits of integration, typically noted as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits.
• Provides a numerical value that signifies the area under the curve between these limits.
• Does not include the constant of integration usually found in indefinite integrals.

To solve a definite integral, follow these steps:

• Evaluate the antiderivative of the function.
• Calculate the difference between the values of the antiderivative at the upper and lower limits.

For our exercise, if it were a definite integral, we would perform these calculations after using the substitution method to find the expression in terms of \( u \). However, as seen in our step-by-step solution, this exercise focused on an indefinite integral.

The indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the given function. Unlike a definite integral, it does not compute the area under a curve between specific bounds.

Key aspects of indefinite integrals include:

• Lack of limits—represented as \( \int f(x) \, dx \).
• Inclusion of a constant of integration, \( C \), because the derivative of a constant is zero, and thus it is included as part of the general solution.

Solving an indefinite integral involves finding a function that, when differentiated, gives the original function. The integral \( \int \frac{(\ln x)^2}{x} \, dx \) from our exercise is evaluated as \( \frac{(\ln x)^3}{3} + C \). The substitution method transforms the integral into \( \int u^2 \, du \), solvable as \( \frac{u^3}{3} + C \), and substituting back gives the solution in terms of \( x \).

Indefinite integrals are foundational to understanding calculus, serving as a step towards solving definite integrals and applications in various fields.