The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ

검증된 단계별 안내

Recognize that the integral is of the form \(\int_0^{\infty} e^{-a\theta} \sin(b\theta) \, d\theta\) where \(a = 1\) and \(b = 1\), but here the integrand is \(2 e^{-\theta} \sin \theta\). Factor out the constant 2 to write the integral as \(2 \int_0^{\infty} e^{-\theta} \sin \theta \, d\theta\).

Recall the standard formula for the integral \(\int_0^{\infty} e^{-p x} \sin(q x) \, dx = \frac{q}{p^2 + q^2}\), valid for \(p > 0\). Here, identify \(p = 1\) and \(q = 1\).

Apply the formula to evaluate \(\int_0^{\infty} e^{-\theta} \sin \theta \, d\theta = \frac{1}{1^2 + 1^2} = \frac{1}{2}\).

Multiply the result by the constant factor 2 that was factored out initially, so the original integral becomes \(2 \times \frac{1}{2}\).

Conclude that the value of the integral is the product found in the previous step, which completes the evaluation without using tables.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, one typically takes limits to handle the infinite bounds, ensuring the integral converges to a finite value.

Integration of Exponential and Trigonometric Functions

Integrals involving products of exponential and trigonometric functions can be solved using integration techniques such as integration by parts or recognizing standard integral forms. These often result in expressions involving both sine and cosine terms.

Convergence of Integrals with Oscillatory Functions

When integrating functions like e^(−θ) sinθ over [0, ∞), the exponential decay ensures the integral converges despite the oscillatory sine term. Understanding how the decay dominates oscillations is key to confirming convergence.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀^∞ 2e^(−θ) sinθ dθ