Problem 9 - Calculus and its applications

Question

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} \frac{d x}{2+x} $$

Step-by-Step Solution

Answer

The integral is divergent.

Step 1: Identify the Form of the Integral

The given integral is an improper integral because it has an infinite limit of integration at the upper end. The integral is:\[\int_{0}^{\infty} \frac{d x}{2+x}\]We need to determine whether this integral converges or diverges, and if it converges, find its value.

Step 2: Rewrite as a Limit

Since the integral is improper due to the upper limit being infinite, we rewrite it using a limit:\[\lim_{b \to \infty} \int_{0}^{b} \frac{d x}{2+x}.\]

Step 3: Evaluate the Integral

We need to find the antiderivative of the integrand \( \frac{1}{2+x} \). The antiderivative is:\[\ln|2+x| + C.\]Now, apply it over the interval \([0, b]\):\[\left. \ln|2+x| \right|_0^b = \ln|2+b| - \ln|2+0|.\]

Step 4: Simplify the Expression

Simplify the expression using logarithmic properties:\[\ln|2+b| - \ln|2| = \ln\left(\frac{2+b}{2}\right).\]

Step 5: Apply the Limit

Now, apply the limit as \( b \to \infty \) to \( \ln\left(\frac{2+b}{2}\right) \):\[\lim_{b \to \infty} \ln\left(\frac{2+b}{2}\right) = \lim_{b \to \infty} \ln\left(1 + \frac{b}{2}\right).\]As \( b \to \infty \), \( \ln\left(1 + \frac{b}{2}\right) \to \infty \) since \( \frac{b}{2} \to \infty \), making the original improper integral divergent.

Key Concepts

Convergence and Divergence

Understanding the concepts of convergence and divergence is essential when dealing with improper integrals. Improper integrals can occur when integration limits extend to infinity, or when the function has infinite discontinuities over the interval of integration. In these situations, there is the question of whether the integral "converges" or "diverges".

• A **convergent** improper integral is one where the area under the curve approaches a finite limit as the upper or lower bounds of integration approach infinity or some form of discontinuity.
• A **divergent** improper integral does not approach a finite limit. Instead, the area under the curve becomes infinitely large.

By determining whether an improper integral converges or diverges, we can ascertain if there is a finite value that represents the integral's solution or not. In our original problem, applying the limit to the logarithmic expression showed that it tends to infinity, which indicates that the integral is divergent. Hence, there is no finite value associated with this improper integral.

Evaluation of Integrals

Evaluating an improper integral involves transforming it into a more manageable form that can be easily interpreted. This often involves rewriting the integral using limits when one of the bounds is infinite or approaching a point of discontinuity.For the integral \[\int_{0}^{\infty} \frac{d x}{2+x},\]we first recognize the need to express it as a limit due to the infinite upper bound:\[\lim_{b \to \infty} \int_{0}^{b} \frac{d x}{2+x}.\]Once rewritten, finding the antiderivative of the integrand \( \frac{1}{2+x} \) becomes crucial for evaluating the integral across the desired domain. The antiderivative turns out to be \( \ln|2+x| + C \), where \( C \) is the constant of integration.Plugging in the limits as the bounds of the integration interval, simplifying through algebraic manipulation, and applying limits helps determine if the integral reaches a finite conclusion or not. This systematic approach facilitates understanding and ensures precision in calculating the integral.

Limits in Calculus

Calculus extensively uses limits to handle problems that involve infinity or indeterminate forms. Limits are foundational tools for evaluating integrals, especially improper ones, where bounds are infinite or discontinuous.The original exercise showcases the power of limits by transforming an infinite integral into a finite expression. By replacing the upper bound of \( \infty \) with an arbitrary point \( b \) and then evaluating the limit as \( b \to \infty \), we transition from an undefined integral into a solvable form:\[\lim_{b \to \infty} \ln\left(\frac{2+b}{2}\right).\]Through limit processes, we assess the behavior of the function near infinity. In this case, the logarithmic expression grows indefinitely because \( \ln\left(1 + \frac{b}{2}\right) \) tends towards infinity as \( b \) rises without bound.Understanding how and when to use limits effectively allows for proper handling of difficult integrals and aids in deciding the eventual convergence or divergence of such expressions.