7–58. Improper integrals Evaluate the following integrals or state that they diverge.
42. ∫ (from 3 to 4) 1/(x-3)³ᐟ² dx

Verified step by step guidance

1. Identify the type of improper integral: The integrand is \( \frac{1}{(x-3)^{3/2}} \), and the lower limit of integration is \( x=3 \), where the denominator becomes zero, indicating a potential vertical asymptote and an improper integral at the lower limit.
2. Rewrite the integral as a limit to handle the improper behavior at \( x=3 \): Express the integral as \( \lim_{t \to 3^+} \int_{t}^{4} \frac{1}{(x-3)^{3/2}} \, dx \). This allows us to evaluate the integral over a domain where the function is defined and then take the limit as \( t \) approaches 3 from the right.
3. Find the antiderivative of the integrand: Use the power rule for integration by rewriting \( (x-3)^{-3/2} \) and integrating with respect to \( x \). Recall that \( \int (x-a)^n \, dx = \frac{(x-a)^{n+1}}{n+1} + C \) for \( n \neq -1 \).
4. Evaluate the definite integral from \( t \) to 4 using the antiderivative found in the previous step.
5. Take the limit as \( t \to 3^+ \) of the evaluated integral to determine if the improper integral converges or diverges.

Key Concepts

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that become unbounded within the interval. To evaluate them, we use limits to approach the problematic point and determine if the integral converges or diverges.

Behavior of Functions Near Singularities

When the integrand has a singularity (like a division by zero) inside the interval, we analyze the function's behavior near that point. Understanding how the function behaves near the singularity helps decide if the integral converges or diverges.

Evaluating Limits for Convergence

To evaluate an improper integral, we replace the problematic bound with a variable and take the limit as it approaches the singularity. If the limit exists and is finite, the integral converges; otherwise, it diverges.