91Ó°ÊÓ

Integration by substitution, also known as the u-substitution method, is a powerful technique used to simplify integrals that are difficult to evaluate in their original form. The main idea is to substitute a part of the integral with a new variable "u" to make it easier to solve.
For example, in the exercise provided, the substitution is performed by letting \( u = x^2 + 1 \). This choice is strategic, as the derivative of \( u \), \( du = 2x \, dx \), directly matches the numerator \( 2x \, dx \) in the integrand.
This substitution transforms the original integral \( \int \frac{2x}{x^2+1} \, dx \) into a much simpler form: \( \int \frac{1}{u} \, du \). This is much easier to integrate since the antiderivative of \( \frac{1}{u} \) is a well-known result, \( \ln |u| + C \).
After integrating, it's important to substitute back the original variable \( x \). This ensures that the solution corresponds to the initial integral, yielding \( \ln |x^2 + 1| + C \).
Key points to remember:

• Choosing the right "u" can greatly simplify the integral.
• The differential \( du \) should match part of the integrand.
• Always substitute back the original variable to complete the solution.

In calculus, limits are essential for understanding the behavior of functions as they approach specific points or infinity. Particularly, when dealing with improper integrals, limits help in evaluating expressions where the bounds extend to infinity or involve points where the function is undefined.
In the exercise, the improper integral \( \int_{2}^{\infty} \frac{2x}{x^2+1} \, dx \) involves an upper limit of infinity. This means the integral cannot be evaluated directly using traditional methods, so we employ the concept of limits.
The first step is to reformulate the integral using limits: \( \lim_{b \to \infty} \int_{2}^{b} \frac{2x}{x^2+1} \, dx \). Here, \( b \) is a variable approaching infinity, replacing the actual infinity in the calculation process. This allows us to evaluate the integral as a "definite" integral first before exploring its behavior as it approaches its boundary.
Remember these points:

• Limits extend the usefulness of integrals to infinite domains.
• By taking limits, we can analyze the convergence or divergence of an integral.
• Make sure to handle infinite limits carefully to understand what happens as the variable increases indefinitely.

Definite integrals compute the accumulation of quantities over a specific interval and provide a numeric value as opposed to indefinite integrals, which yield functions as answers. They play a crucial role in determining the convergence of an improper integral when a function is integrated over a range including infinity.
In the scenario given, after using integration by substitution, the task involves calculating the definite integral \( \lim_{b \to \infty} \left( \ln|b^2+1| - \ln|5| \right) \). This expression results from substituting and then applying the limits.
After evaluating the integral for the finite range \( [2, b] \), applying limits suggests evaluating this expression as \( b \) approaches infinity. Simplification results in identifying that \( \ln(b^2+1) \) diverges as \( b \) grows larger, because a logarithm function continually increases with its input, highlighting the divergence of the integral.
Important considerations include:

• To evaluate definite integrals, handle the bounds carefully.
• Improper integrals often require turning their evaluation into a limit problem.
• Pay attention to whether the integral converges or diverges.
• Understanding the behavior of functions such as logarithms is crucial in discerning divergence.