To determine the definite integral of the function [tex]\( f(x) = \frac{12 (\ln x)^3}{x} \)[/tex] over the interval [tex]\( [1, 3] \)[/tex], we follow these steps:
1. Understand the function:
[tex]\[
f(x) = \frac{12 (\ln x)^3}{x}
\][/tex]
This function involves the natural logarithm of [tex]\( x \)[/tex] raised to the third power, multiplied by 12, and then divided by [tex]\( x \)[/tex].
2. Set up the definite integral:
We need to integrate [tex]\( f(x) \)[/tex] from [tex]\( x = 1 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
\int_{1}^{3} \frac{12 (\ln x)^3}{x} \, dx
\][/tex]
3. Evaluate the integral:
To evaluate the definite integral, we need to proceed through an integration process which typically might involve techniques such as integration by parts or substitution. In this case, rather than going through those steps manually, we acknowledge that the integral of the given function over the specified interval yields:
[tex]\[
\int_{1}^{3} \frac{12 (\ln x)^3}{x} \, dx = 4.370177382019715
\][/tex]
4. Interpret the result:
This numerical value represents the area under the curve [tex]\( f(x) = \frac{12 (\ln x)^3}{x} \)[/tex] from [tex]\( x = 1 \)[/tex] to [tex]\( x = 3 \)[/tex].
Thus, the value of the definite integral is approximately [tex]\( 4.370177382019715 \)[/tex].
