Final Answer:
The given integral ∫₀¹ ln(x) dx is divergent.
Step-by-step explanation:
The integral ∫₀¹ ln(x) dx represents the area under the curve of the natural logarithm function from x = 0 to x = 1. To analyze its convergence, we need to assess whether the integral converges or diverges. The integral diverges due to the singularity at x = 0 in the natural logarithm function. As x approaches 0, ln(x) approaches negative infinity. This behavior causes the integral to diverge since the function is unbounded near x = 0.
Mathematically, when evaluating the integral ∫₀¹ ln(x) dx directly from the definition, it can be represented as lim┬(a→0⁺) ∫┬(a)¹ ln(x) dx. Applying this limit, as a approaches 0 from the positive side, the logarithmic function becomes unbounded, resulting in the integral approaching negative infinity. Hence, the integral diverges, signifying that the area under the curve of ln(x) from 0 to 1 extends indefinitely and does not converge to a finite value.
Therefore, based on the nature of the natural logarithm function where ln(x) approaches negative infinity as x tends to 0^+, the integral ∫₀¹ ln(x) dx is divergent, indicating that the area bounded by the curve and the x-axis is infinite, and consequently, the definite integral cannot be computed as a finite value.