Step 1: Understanding the Concept:
The integrand involves the greatest integer function (or floor function) denoted by $[x]$. This function outputs a constant integer value for all $x$ within a given interval between two consecutive integers. To evaluate the definite integral, we must partition the total interval of integration into smaller sub-intervals where the value of $[x]$ remains constant.
Step 2: Key Formula or Approach:
1. Definite integral additivity property over intervals: $\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$.
2. Definition of $[x]$: $[x] = n$ for $n \le x<n+1$, where $n$ is an integer.
Step 3: Detailed Explanation:
We are given the definite integral $I = \int_1^4 \log[x] dx$.
The interval of integration is from $1$ to $4$. The value of $[x]$ changes at integer boundaries. The integer points within this range are $2$ and $3$.
We split the integral at these points:
\[ I = \int_1^2 \log[x] dx + \int_2^3 \log[x] dx + \int_3^4 \log[x] dx \]
Now determine the constant value of $[x]$ in each specific sub-interval:
- In the interval $[1, 2)$, the value of $[x]$ is $1$. The integrand becomes $\log(1) = 0$.
- In the interval $[2, 3)$, the value of $[x]$ is $2$. The integrand becomes $\log(2)$.
- In the interval $[3, 4)$, the value of $[x]$ is $3$. The integrand becomes $\log(3)$.
Note: the value at isolated boundaries (like exactly at $x=4$) doesn't change the area under the curve.
Substitute these values back into the respective integrals:
\[ I = \int_1^2 \log(1) dx + \int_2^3 \log(2) dx + \int_3^4 \log(3) dx \]
Since $\log(1) = 0$, the first term vanishes. The remaining logs are constants and can be pulled out of the integrals:
\[ I = 0 + \log(2) \int_2^3 1 dx + \log(3) \int_3^4 1 dx \]
Evaluate the simple definite integrals:
\[ I = \log(2) \cdot [x]_2^3 + \log(3) \cdot [x]_3^4 \]
\[ I = \log(2) \cdot (3 - 2) + \log(3) \cdot (4 - 3) \]
\[ I = \log(2) \cdot 1 + \log(3) \cdot 1 \]
\[ I = \log 2 + \log 3 \]
Apply the logarithmic addition property $\log a + \log b = \log(ab)$:
\[ I = \log(2 \cdot 3) = \log 6 \]
Step 4: Final Answer:
The value of the integral is $\log 6$.