Option: a is the correct answer.
a. (0, ∞)
We are given a logarithmic function f(x) as:
[tex]f(x)=\ln x[/tex]
We know that the logarithmic function is defined for all the real values strictly greater than 0 i.e. x>0.
i.e. the function is defined for all positive real numbers.
i.e. the domain of the function f(x) is: (0,∞).
Also, we know that the function f(x) is integrable in it's domain and the integration is calculated by using the integration by parts.
i.e.
[tex]\int\limits {\ln x} \, dx=\int\limits {1\cdot \ln x} \, dx\\ \\i.e.\\\\\int\limits {\ln x} \, dx=\ln x\cdot \int\limits {1} \, dx-\int\limits {\dfrac{d}{dx}\ln x} \cdot \int\limits {1} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=\ln x\cdot x-\int\limits {\dfrac{1}{x}\cdot x} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=x\cdot \ln x-\int\limits {1} \, dx\\\\i.e.\\\\\int\limits {\ln x} \, dx=x\cdot \ln x-x[/tex]
Hence, the answer is: Option: a
