Final answer:
The integral of ln(x) / x from 0 to 1 is not defined because ln(x) is not defined at x = 0. The value of this integral is associated with the Euler-Mascheroni constant, and numerical methods or limits are used for its evaluation.
Step-by-step explanation:
We want to evaluate the integral of ln(x) / x from 0 to 1. This is a special integral, often represented as the integral of ln(x) dx over the bounds from 0 to 1. However, the direct computation of this integral over the given bounds is not possible because ln(x) is not defined at x = 0, and the indefinite integral of ln(x)/x is a well-known function called the logarithmic integral, which has no elementary form.
The integral ∫ (ln(x) / x) dx from 0 to 1, often denoted as Li(1), is famously known to be equal to -γ, where γ is the Euler-Mascheroni constant, approximately 0.57722. However, in practice, numerical methods or approximation techniques are employed to evaluate such integrals. To directly evaluate the integral using basic calculus tools, one must consider a proper limit process or use series expansion.
One common approach is to evaluate the integral from ε to 1, where ε > 0, and take the limit as ε approaches 0+. Using integration by parts or series expansions and then taking the limit can yield the result indirectly. Nonetheless, this question is theoretically more complex than it might initially appear due to the singularity at x = 0.