7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv

Verified step by step guidance

1. Identify that the integral \( \int_1^{\infty} \frac{1}{v(v+1)} \, dv \) is an improper integral because the upper limit is infinite.
2. Rewrite the integral as a limit: \( \lim_{t \to \infty} \int_1^{t} \frac{1}{v(v+1)} \, dv \). This allows us to handle the infinite limit properly.
3. Use partial fraction decomposition to simplify the integrand: express \( \frac{1}{v(v+1)} \) as \( \frac{A}{v} + \frac{B}{v+1} \) and solve for constants \( A \) and \( B \).
4. Integrate the decomposed expression term-by-term over \( v \) from 1 to \( t \), resulting in logarithmic functions.
5. Evaluate the definite integral by substituting the limits 1 and \( t \), then take the limit as \( t \to \infty \) to determine if the integral converges or diverges.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence depends on whether this limit exists finitely.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational function into simpler fractions that are easier to integrate. For example, 1/[v(v + 1)] can be rewritten as A/v + B/(v + 1), allowing straightforward integration of each term separately.

Evaluating Limits of Integrals

When dealing with improper integrals, after integrating the function, you must evaluate the limit of the antiderivative as the variable approaches infinity. This step determines if the integral converges to a finite value or diverges.