Problem 55 - Thomas Calculus

Question

In Exercises \(35-64\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$ \int_{\pi}^{\infty} \frac{2+\cos x}{x} d x $$

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Simplify the Integral

Consider the integral \( \int_{\pi}^{\infty} \frac{2+\cos x}{x} \, dx \). We can split this integral into two parts: \( \int_{\pi}^{\infty} \frac{2}{x} \, dx \) and \( \int_{\pi}^{\infty} \frac{\cos x}{x} \, dx \). This separation will help in analyzing convergence.

Step 2: Evaluate Convergence of \( \int_{\pi}^{\infty} \frac{2}{x} \, dx \)

The integral \( \int_{\pi}^{\infty} \frac{2}{x} \, dx \) is similar to the \( p \)-integral \( \int_{\pi}^{\infty} \frac{1}{x} \, dx \), which diverges because \( p = 1 \leq 1 \). Therefore, \( \int_{\pi}^{\infty} \frac{2}{x} \, dx \) also diverges.

Step 3: Analyze \( \int_{\pi}^{\infty} \frac{\cos x}{x} \, dx \) using Comparison Tests

Consider the comparison \( 0 \leq \left| \frac{\cos x}{x} \right| \leq \frac{1}{x} \). The integral \( \int_{\pi}^{\infty} \frac{1}{x} \, dx \) diverges, indicating that the behavior of \( \frac{\cos x}{x} \) is not sufficient to cancel out the divergence of \( \frac{2}{x} \). Thus, it does not affect the divergence of the entire original integral.

Step 4: Combine Results for Final Conclusion

The integral \( \int_{\pi}^{\infty} \frac{2+\cos x}{x} \, dx \) diverges because \( \int_{\pi}^{\infty} \frac{2}{x} \, dx \) diverges and the cosine term \( \frac{\cos x}{x} \) does not converge sufficiently to alter the divergence of the overall integral.

Key Concepts

Direct Comparison Test

The Direct Comparison Test is a handy tool for assessing the convergence or divergence of improper integrals. It involves comparing an unknown integral to another integral whose behavior is well understood. The basic idea is to establish if the function inside the integral is always less than or greater than another function for which convergence behavior is known.

To apply the Direct Comparison Test, choose a function that is simpler and known to converge or diverge over the given interval. If the function inside your integral is always less than a convergent comparator, then the original integral converges. Conversely, if it's always greater than a divergent comparator, then the original integral diverges.

This method was used intuitively in the solution by comparing parts of the original function to a simpler divergent function, which helped conclude that the original integral also diverges due to its greater parts.

• Choose an appropriate function for comparison.
• Ensure your comparator function's convergence properties are well known.
• Compare over the applicable range of integration to use the test effectively.

Limit Comparison Test

When the Direct Comparison Test becomes cumbersome, especially when the expressions don't neatly allow for straightforward bounding, the Limit Comparison Test comes in as a strong tool.
This test involves taking the limit of the ratio of the function of the integral in question and a comparator function. The comparator should, ideally, be simple and have a known convergence behavior.

The Limit Comparison Test follows these steps:

• Identify a simple function (b(x)) whose convergence behavior is understood.
• Compute: \[ L = \lim_{x \to \infty} \frac{f(x)}{b(x)} \]
• If \(L\) is finite and positive, both integrals either converge or diverge together.

In the original exercise, even though this specific test wasn't needed, understanding how to leverage it with limits can offer clarity, especially when the integrand behaves in more complex ways not easily captured by Direct Comparison.

Improper Integrals

Improper integrals extend the notion of definite integrals to unbounded intervals or functions with undefined independence points.
These integrals type often deal with functions integrated over infinite intervals or within points at which they escalate to infinity.
Integration techniques such as the Direct and Limit Comparison Tests fit perfectly into analysis of these integrals due to possible divergence.

An integral of the type mentioned in the original exercise, \( \int_{a}^{igsqcup} f(x) \, dx \), requires careful handling:

• Identify intervals over which the function behaves properly.
• If the upper limit is infinite (like our exercise), evaluate if it can be simplified involving known divergent or convergent integrals.
• Proceed with splitting or approximating technique to clarify the behavior of the original integral over limits.

Recognizing an improper integral is crucial. This recognition allows utilizing appropriate convergence tests, as improper behavior leads to informative divergence patterns like those needed to solve our specific integral problem.