Find the Antiderivative f(x)=9/x

Solution:

Step 1:

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

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

Step 2:

Write down the integral that needs to be solved.

$$F(x) = \int \frac{9}{x} \, dx$$

Step 3:

Extract the constant $9$ from the integral as it is not dependent on $x$.

$$9 \int \frac{1}{x} \, dx$$

Step 4:

Integrate $\frac{1}{x}$ with respect to $x$, which is known to be the natural logarithm of the absolute value of $x$.

$$9 \left( \ln |x| + C \right)$$

Step 5:

Express the result in a simplified form.

$$9 \ln |x| + C$$

Step 6:

Conclude with the antiderivative of the given function $f(x) = \frac{9}{x}$.

$$F(x) = 9 \ln |x| + C$$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the operation of differentiation. The antiderivative of a function $f(x)$ is another function $F(x)$ whose derivative is $f(x)$, denoted as:

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

When integrating a function, constants can be factored out of the integral, simplifying the process. The integral of $\frac{1}{x}$ is a fundamental integral that results in the natural logarithm of the absolute value of $x$, expressed as:

$$\int \frac{1}{x} \, dx = \ln |x| + C$$

Here, $C$ represents the constant of integration, which is included because the antiderivative of a function is not unique; any function differing by a constant has the same derivative.

The absolute value is used in the natural logarithm to ensure the domain of the logarithm function is satisfied, as the logarithm is only defined for positive real numbers. Therefore, $|x|$ ensures that the input to the logarithm function is always positive.

In summary, the relevant knowledge points for solving this problem are:

1. Understanding the concept of antiderivatives and indefinite integrals.

2. Applying the constant multiple rule in integration.

3. Knowing the integral of $\frac{1}{x}$.

4. Recognizing the need for a constant of integration.

5. Understanding the domain of the natural logarithm function and the use of absolute values.