To solve the definite integral [tex]\(\int_{12}^1 f(x) \, dx\)[/tex], we need to follow these detailed steps:
1. Identify the Function [tex]\( f(x) \)[/tex]:
To compute the integral of a function, we must know what [tex]\( f(x) \)[/tex] is. However, in this problem [tex]\( f(x) \)[/tex] is not provided. Without knowing the specific function [tex]\( f(x) \)[/tex], it is impossible to proceed with the integration process.
2. Evaluate the Integral with Limits:
If we had the function [tex]\( f(x) \)[/tex], the next step would be to find the antiderivative [tex]\( F(x) \)[/tex] of the function [tex]\( f(x) \)[/tex]. The fundamental theorem of calculus states that if [tex]\( F(x) \)[/tex] is an antiderivative of [tex]\( f(x) \)[/tex], then:
[tex]\[
\int_{a}^b f(x) \, dx = F(b) - F(a)
\][/tex]
In our case, the limits of integration are from 12 to 1.
3. Calculate Definite Integral:
Plugging in the upper and lower limits into the antiderivative, the definite integral would be evaluated as:
[tex]\[
\int_{12}^1 f(x) \, dx = F(1) - F(12)
\][/tex]
However, as previously mentioned, since the function [tex]\( f(x) \)[/tex] is not specified, we cannot find [tex]\( F(x) \)[/tex] and thus cannot perform this step.
4. Conclusion:
The integral [tex]\(\int_{12}^1 f(x) \, dx\)[/tex] is incomplete and cannot be computed without the given function [tex]\( f(x) \)[/tex]. Therefore, based on the information provided, it's not possible to determine the value of the integral.
In summary, the computation of the integral [tex]\(\int_{12}^1 f(x) \, dx\)[/tex] cannot be completed because the function [tex]\( f(x) \)[/tex] is missing. We need the explicit expression of [tex]\( f(x) \)[/tex] to proceed with the integration.
