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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.8.66
Chapter 8, Problem 8.8.66

In Exercises 35–68, 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.
∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)

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First, identify the integral to be tested for convergence: \(\displaystyle \int_1^{\infty} \frac{1}{e^x - 2^x} \, dx\).
Examine the behavior of the integrand as \(x \to \infty\). Since both \(e^x\) and \$2^x$ grow exponentially, compare their growth rates to understand the dominant term in the denominator.
Determine which function grows faster: \(e^x\) or \$2^x\(. Recall that \(e \approx 2.718\), so \)e^x\( grows faster than \)2^x\(. Therefore, for large \)x\(, \)e^x - 2^x\( behaves approximately like \)e^x$.
Use the Direct Comparison Test by comparing the integrand to a simpler function that behaves like \(\frac{1}{e^x}\) for large \(x\). Since \(\frac{1}{e^x}\) is a convergent integral on \([1, \infty)\), this comparison can help determine convergence.
Alternatively, apply the Limit Comparison Test by computing the limit \(\displaystyle \lim_{x \to \infty} \frac{\frac{1}{e^x - 2^x}}{\frac{1}{e^x}}\) to confirm if the integrals behave similarly at infinity, which will indicate whether the original integral converges or diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Improper Integrals and Convergence

Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To determine convergence, we evaluate the limit of the integral as the upper bound approaches infinity. If this limit exists and is finite, the integral converges; otherwise, it diverges.
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Improper Integrals: Infinite Intervals

Direct Comparison Test

The Direct Comparison Test compares the given integral's integrand to a simpler function whose convergence behavior is known. If the integrand is less than or equal to a convergent function, the integral converges; if it is greater than or equal to a divergent function, the integral diverges.
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Limit Comparison Test

The Limit Comparison Test uses the limit of the ratio of the given integrand to a known function as x approaches infinity. If the limit is a positive finite number, both integrals either converge or diverge together, helping to determine the behavior of complex integrals.
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Related Practice
Textbook Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ √(x - x²) / x dx

Textbook Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x (7x + 5)^(3/2) dx

Textbook Question

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x dx) / (25 + 4x²)

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Textbook Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x⁵ e³ˣ dx

Textbook Question

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ ln(x + x²) dx