Find the Antiderivative f(x)=9

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$$F(x) = \int f(x) \, dx$$

Step 2:

Write down the integral that needs to be solved.

$$F(x) = \int 9 \, dx$$

Step 3:

Utilize the rule for integrating a constant.

$$F(x) = 9x + C$$

Step 4:

Conclude with the antiderivative of $f(x) = 9$.

$$F(x) = 9x + C$$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. Here are the relevant knowledge points:

1. Indefinite Integral: The indefinite integral of a function $f(x)$, denoted as $\int f(x) \, dx$, represents the family of all antiderivatives of $f(x)$.

2. Constant Rule: When integrating a constant $a$, the antiderivative is $ax + C$, where $C$ is the constant of integration. This rule is applied because the derivative of $ax$ with respect to $x$ is $a$.

3. Constant of Integration: The constant $C$ is included in the antiderivative because the derivative of a constant is zero, which means any constant could have been present in the original function before differentiation.

4. Integration Process: To integrate a function, one typically applies known integration rules and techniques that correspond to the reverse of differentiation rules, such as the power rule, product rule, chain rule, etc.

In the given problem, the function to integrate is a constant, $f(x) = 9$. According to the constant rule, the antiderivative is $9x + C$, where $C$ is the constant of integration. This reflects the fact that the derivative of $9x$ is $9$, and any constant could have been present in the original function.