The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999

검증된 단계별 안내

Identify the integral to be evaluated: \(\int_0^1 \frac{dr}{r^{0.999}}\).

Rewrite the integrand using exponent rules: \(\frac{1}{r^{0.999}} = r^{-0.999}\), so the integral becomes \(\int_0^1 r^{-0.999} \, dr\).

Recall the power rule for integration: For \(\int r^n \, dr\), where \(n \neq -1\), the antiderivative is \(\frac{r^{n+1}}{n+1} + C\).

Apply the power rule to the integral: \(\int_0^1 r^{-0.999} \, dr = \left[ \frac{r^{-0.999 + 1}}{-0.999 + 1} \right]_0^1 = \left[ \frac{r^{0.001}}{0.001} \right]_0^1\).

Evaluate the definite integral by substituting the limits: calculate \(\frac{1^{0.001}}{0.001} - \frac{0^{0.001}}{0.001}\), noting the behavior of \(r^{0.001}\) as \(r\) approaches 0.

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주요 개념

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

Improper Integrals

Improper integrals involve integrands that are unbounded or have infinite limits. When the integrand approaches infinity at a boundary, the integral is evaluated as a limit to determine convergence or divergence.

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. This rule is essential for integrating functions with variable exponents, allowing direct evaluation of integrals involving powers of the variable.

Convergence of Integrals with Singularities

When the integrand has a singularity (like r^(-0.999) near zero), determining if the integral converges requires analyzing the behavior near the singularity. If the integral's limit exists and is finite, the integral converges.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀¹ dr / r^0.999