91Ó°ÊÓ

Integration is a fundamental concept in calculus. It is the process of finding a function, known as the antiderivative, whose derivative is the given function. In this exercise, we have the derivative of a function, \(f'(x) = x - 5\), and we need to find the original function, \(f(x)\).
To integrate \(f'(x) = x - 5\), we split the integral into two parts:
\[\int (x - 5) \, dx = \int x \, dx - \int 5 \, dx\]
This becomes:
\[\int x \, dx - 5 \int \, dx\]
\[\frac{x^2}{2} - 5x + C\]
Where \(C\) is the constant of integration, a number that can be any value. Since differentiation eliminates constants, integration introduces one.
Thus, we get the integrated function:
\[f(x) = \frac{x^2}{2} - 5x + C\]
This gives us the general form of the function before specifying any initial conditions.

Initial conditions help determine the specific function within a family of functions. Here, the problem provides an initial condition \(f(1) = 6\).
This means that when \(x = 1\), the function \(f(x)\) equals 6.
We use this to solve for the constant \(C\) in our previously integrated result.
Substitute \(x = 1\) and \(f(1) = 6\) into \(f(x) = \frac{x^2}{2} - 5x + C\)

\[f(1) = \frac{1^2}{2} - 5(1) + C\]
\[6 = \frac{1}{2} - 5 + C\]
This simplifies to:
\[-4.5 + C = 6\]
Using the initial condition ensures that our function \(f(x)\) passes through the specific point (1, 6).

After applying the initial condition, we can solve for the constant \(C\). Given the equation:
\[-4.5 + C = 6\]
Rearrange to find \(C\):
\[C = 6 + 4.5\]
\[C = 10.5\]
Thus, we find the specific value of \(C\) that ensures the initial condition is met.
We then substitute it back into the integrated function\[f(x) = \frac{x^2}{2} - 5x + 10.5\]
We now have the complete solution to the original problem. The derived function correctly represents all aspects including the slope and initial condition given in the problem.
Understanding how to solve for constants in integration problems allows us to find exact solutions rather than general ones.