7–58. Improper integrals Evaluate the following integrals or state that they diverge.
53. ∫ (from 0 to 1) ln x dx

Verified step by step guidance

1. Recognize that the integral \( \int_0^1 \ln x \, dx \) is an improper integral because \( \ln x \) approaches \(-\infty\) as \( x \) approaches 0 from the right.
2. Rewrite the integral as a limit to handle the improper behavior at 0: \[ \int_0^1 \ln x \, dx = \lim_{t \to 0^+} \int_t^1 \ln x \, dx \].
3. Use integration by parts to evaluate \( \int \ln x \, dx \). Let \( u = \ln x \) and \( dv = dx \), then \( du = \frac{1}{x} dx \) and \( v = x \).
4. Apply the integration by parts formula: \[ \int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 \, dx = x \ln x - x + C \].
5. Evaluate the definite integral using the antiderivative: \[ \int_t^1 \ln x \, dx = [x \ln x - x]_t^1 = (1 \cdot \ln 1 - 1) - (t \ln t - t) \]. Then take the limit as \( t \to 0^+ \) to determine if the integral converges or diverges.

Key Concepts

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that approach infinity within the interval. To evaluate them, we use limits to handle points where the function is undefined or unbounded, determining if the integral converges or diverges.

Natural Logarithm Function Behavior near Zero

The natural logarithm function, ln(x), approaches negative infinity as x approaches zero from the right. This behavior creates an improper integral at the lower limit, requiring careful limit evaluation to determine convergence.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, useful for integrating products of functions. It transforms the integral into simpler parts, often applied when integrating functions like ln(x) that are difficult to integrate directly.