Question

What is the value of \[\int_0^1 {\log [x]dx} \]?
A. 0
B. 1
C. log1
D. None of these

Answer 1

Hint: In solving the above question, first we will find the integration of the function using integration formulas and, then substitute the upper limit and lower limit values of \[x\] given and after simplifying the result we will get the desired value.

  Formula used:
We will use integration formula i.e., \[\int {\log x = x\left( {\log x - 1} \right)} \], and we will also use the fact that \[\log 0\] is undefined.

  Complete Step-by- Step Solution:
Given \[\int_0^1 {\log [x]dx} \]
Now we will apply integration formula, we will get,
\[\int_0^1 {\log [x]dx} = \left[ {x\left( {\log x - 1} \right)} \right]_0^1\]
Now we will apply the upper limit value and lower limit value of the function as per given, we will get,
\[\int_0^1 {\log [x]dx} = \left[ {1\left( {\log 1 - 1} \right)} \right] - \left[ {0\left( {\log 0 - 1} \right)} \right]\]
Now we will us fact that the value of \[\log 0\] is undefined and the value of \[\log 1 = 0\], then we will get,
\[\int_0^1 {\log [x]dx} = \left[ {1\left( {0 - 1} \right)} \right] - \]undefined,
Now we will simplify the right hand side of the equation, we will get,
\[\int_0^1 {\log [x]dx} = \]undefined.
The correct option is D.

Note: Many student do not aware about the value of \[\log 0\]. The value of \[\log 0\] is undefined. Also the sum of a real number with undefined is always undefined.