Evaluate the improper integrals in Exercises 53–62.
∫ from 3 to ∞ of (2 / (u² − 2u)) du

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First, recognize that the integral is improper because the upper limit is infinity. So, rewrite the integral as a limit: \(\displaystyle \int_{3}^{\infty} \frac{2}{u^{2} - 2u} \, du = \lim_{t \to \infty} \int_{3}^{t} \frac{2}{u^{2} - 2u} \, du\).

Next, factor the denominator to simplify the integrand: \(u^{2} - 2u = u(u - 2)\). So the integrand becomes \(\frac{2}{u(u - 2)}\).

Use partial fraction decomposition to express \(\frac{2}{u(u - 2)}\) as \(\frac{A}{u} + \frac{B}{u - 2}\). Set up the equation \(2 = A(u - 2) + Bu\) and solve for constants \(A\) and \(B\).

After finding \(A\) and \(B\), rewrite the integral as the sum of two simpler integrals: \(\int \frac{A}{u} \, du + \int \frac{B}{u - 2} \, du\). Integrate each term separately to get logarithmic functions.

Finally, substitute the limits from 3 to \(t\), then take the limit as \(t\) approaches infinity. Evaluate the resulting expression to determine if the improper integral converges or diverges.

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주요 개념

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Improper Integrals

Improper integrals involve integration over an infinite interval or where the integrand has an infinite discontinuity. To evaluate them, the integral is expressed as a limit, such as taking the limit as the upper bound approaches infinity. This approach ensures the integral converges to a finite value or diverges.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. For example, a function like 2/(u² − 2u) can be rewritten as a sum of fractions with linear denominators, facilitating straightforward integration of each term.

Limits and Convergence of Integrals

Evaluating improper integrals requires analyzing the limit of the integral as the variable approaches infinity. Determining whether this limit exists (converges) or not (diverges) is crucial. Convergence means the area under the curve is finite, allowing the integral to be assigned a value.

Evaluate the improper integrals in Exercises 53–62.∫ from 3 to ∞ of (2 / (u² − 2u)) du