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svetlana [45]
2 years ago
7

Determine the interval on which f(x) = ln(x) is integrable.

Mathematics
2 answers:
Juliette [100K]2 years ago
8 0
<span>a. (0, ∞)
....................................</span>
PilotLPTM [1.2K]2 years ago
3 0
<h2>Answer:</h2>

  Option: a is the correct answer.

                  a.      (0, ∞)

<h2>Step-by-step explanation:</h2>

We are given a logarithmic function f(x) as:

f(x)=\ln x

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.

\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

Hence, the answer is: Option: a

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