Problem 18 - Calculus Concepts

Question

For Activities 17 through \(22,\) use algebraic manipulation or integration by substitution as well as limits to evaluate the improper integral. $$ \int_{2}^{\infty} \frac{x^{2}+1}{2 x} d x $$

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Simplify the Integrand

Rewrite the integrand \( \frac{x^2 + 1}{2x} \) by separating the fraction: \( \frac{x^2}{2x} + \frac{1}{2x} = \frac{x}{2} + \frac{1}{2x} \). This simplifies the expression, which makes the integration manageable.

Step 2: Integrate Each Term

Now, integrate the expression term by term. The integral becomes: \[ \int \left( \frac{x}{2} + \frac{1}{2x} \right) \, dx = \int \frac{x}{2} \, dx + \int \frac{1}{2x} \, dx. \] The integral of \( \frac{x}{2} \) with respect to \( x \) is \( \frac{x^2}{4} \), and the integral of \( \frac{1}{2x} \) with respect to \( x \) is \( \frac{1}{2} \ln|x| \).

Step 3: Apply Limits of Integration

Now apply the limits from 2 to \( \infty \) for the improper integral. We rewrite the integral with the upper limit as \( t \) and then take the limit as \( t \to \infty \). We have: \[ \int_{2}^{t} \left( \frac{x^2}{4} + \frac{1}{2} \ln|x| \right) \, dx = \left[ \frac{x^2}{4} + \frac{1}{2} \ln |x| \right]_{2}^{t}. \]

Step 4: Evaluate the Integral at the Limits

Evaluate the expression at \( t \) and 2: \[ \lim_{t \to \infty} \left( \frac{t^2}{4} + \frac{1}{2} \ln|t| \right) - \left( \frac{2^2}{4} + \frac{1}{2} \ln|2| \right). \] The term \( \frac{t^2}{4} \) approaches \( \infty \) as \( t \to \infty \), which indicates that the integral diverges.

Key Concepts

Integration by Substitution

Integration by substitution is a fundamental technique in calculus that simplifies the process of finding antiderivatives for complex integrands. The goal is to substitute part of the integrand with a new variable, making the integration simpler. To perform integration by substitution:

• Identify a portion of the integrand that can be substituted with a single variable, typically "u".
• Differentiate this part to find "du", which will replace "dx" in the integral.
• Rewrite the integrand in terms of "u" and "du".
• Integrate with respect to "u".
• Finally, substitute back the original variable to complete the integration.

This method is particularly helpful when dealing with integrals involving compositions of functions or complicated polynomials. It transforms an otherwise daunting task into a more manageable one by reducing the complexity of the integral. In our given example, integration by substitution could be considered, but due to the polynomial expressions in the numerator and denominator, algebraic manipulation proves more convenient.

Algebraic Manipulation

Algebraic manipulation involves rewriting an expression to simplify it, often using basic algebra rules such as factoring, distributing, or breaking down fractions. This technique turns a complex expression into more manageable ones, which is extremely useful in calculus while evaluating integrals.To simplify \[\frac{x^2 + 1}{2x}\]through algebraic manipulation, we separate the fraction into two parts:

• \(\frac{x^2}{2x} = \frac{x}{2}\), simplifying based on the common factor.
• \(\frac{1}{2x}\) which remains unchanged for immediate integration.

Thus, the original integrand is rearranged as \[\frac{x}{2} + \frac{1}{2x}\],creating a simpler form, which can be integrated easily.This simplification aids immensely in handling improper integrals, as it breaks down complex expressions into summands that can be integrated term by term. In integrals involving rational expressions, algebraic manipulation often provides clarity by exposing simpler patterns for integration.

Limits in Calculus

Limits in calculus play a crucial role, especially when dealing with improper integrals. An improper integral is one where either the interval of integration is infinite, or the function becomes unbounded within the interval. To handle such integrals, limits help by providing a way to evaluate them meaningfully.The process involves substituting a boundary approaching infinity with a finite symbol, like "\(t\)", evaluating the integral with fixed limits, and then taking the limit of the result as "\(t\)" approaches infinity:

• Replace the upper limit \( \infty \) with \( t \).
• Integrate from the lower limit to \( t \).
• Take the limit as \( t\to\infty \).

For our improper integral \[\int_{2}^{\infty} \left( \frac{x}{2} + \frac{1}{2x} \right) \, dx\], we substituted the upper limit with \( t \).As \( t \rightarrow \infty \), the expression approaches infinity, indicating divergence. Limits underscore the behavior of functions at extreme boundaries, providing insight into the convergence or divergence of an integral, hence allowing for a proper evaluation even when a function behaves erratically at the endpoints.