7–58. Improper integrals Evaluate the following integrals or state that they diverge.
44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy

검증된 단계별 안내

Identify the integral to evaluate: \(\int_0^{\ln 3} \frac{e^y}{(e^y - 1)^{7/3}} \, dy\).

Check the behavior of the integrand at the limits of integration, especially near \(y=0\) and \(y=\ln 3\), to determine if the integral is improper. Note that at \(y=0\), \(e^y - 1 = 0\), which may cause a singularity.

To analyze the singularity at \(y=0\), perform a substitution or use a limit approach. For example, let \(x = e^y - 1\), so when \(y \to 0\), \(x \to 0\). Express the integrand in terms of \(x\) and examine the limit as \(x \to 0\).

Rewrite the integral using the substitution \(x = e^y - 1\), which implies \(dx = e^y dy = (x + 1) dy\), so \(dy = \frac{dx}{x + 1}\). Substitute into the integral and adjust the limits accordingly.

Evaluate the integral or determine convergence by analyzing the behavior of the integrand near the singularity and then proceed with integration techniques such as substitution or integration by parts if necessary.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Improper Integrals

Improper integrals involve integration over an unbounded interval or integrands with unbounded behavior within the interval. To evaluate them, one often takes limits approaching the problematic points to determine convergence or divergence.

Behavior of the Integrand Near Singularities

Analyzing the integrand's behavior near points where it may become infinite or undefined is crucial. For example, if the denominator approaches zero, the integrand may have a singularity, affecting convergence of the integral.

Substitution and Simplification Techniques

Using substitution can simplify the integral, especially when the integrand involves composite functions like exponentials. Simplifying the expression helps in identifying limits and evaluating the integral more easily.

7–58. Improper integrals Evaluate the following integrals or state that they diverge.44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy