Find the Antiderivative f(x)=-12x

Solution:

Step 1:

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

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

Step 2:

Prepare the integral for computation.

$$F(x)=\int -12xdx$$

Step 3:

Extract the constant $-12$ from the integral as it does not depend on $x$.

$$-12\int xdx$$

Step 4:

Apply the Power Rule for integration to find the integral of $x$.

$$-12(\frac{x^2}{2}+C)$$

Step 5:

Proceed to simplify the expression.

Step 5.1:

Rewrite the expression by distributing the constant $-12$.

$$-12\left(\frac{1}{2}\right)x^2+C$$

Step 5.2:

Perform the simplification.

Step 5.2.1:

Multiply $-12$ by $\frac{1}{2}$.

$$\frac{-12}{2}{x}^2+C$$

Step 5.2.2:

Reduce the fraction by eliminating common factors.

Step 5.2.2.1:

Separate the factor of $2$ from $-12$.

$$\frac{2\cdot (-6)}{2}{x}^2+C$$

Step 5.2.2.2:

Eliminate the common factors.

Step 5.2.2.2.1:

Isolate the factor of $2$ in the denominator.

$$\frac{2\cdot (-6)}{2\cdot 1}{x}^2+C$$

Step 5.2.2.2.2:

Cancel out the common factor of $2$.

$$\frac{\cancel{2}\cdot (-6)}{\cancel{2}\cdot 1}{x}^2+C$$

Step 5.2.2.2.3:

Rewrite the simplified expression.

$$\frac{-6}{1}{x}^2+C$$

Step 5.2.2.2.4:

Divide $-6$ by $1$.

$$-6x^2+C$$

Step 6:

Conclude with the antiderivative of the function $f(x)=-12x$.

$$F(x)=-6x^2+C$$

Knowledge Notes:

To solve for the antiderivative of a function, one must be familiar with the process of integration, which is essentially finding the function whose derivative is the given function. Here are the relevant knowledge points and explanations:

1. Indefinite Integral: The antiderivative of a function is also known as the indefinite integral. It is represented by the integral sign followed by the function and the differential, e.g., $\int f(x)dx$.

2. Constant Factor Rule: When integrating a function multiplied by a constant, the constant can be pulled out of the integral, e.g., $\int cf(x)dx=c\int f(x)dx$.

3. Power Rule for Integration: This rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}+C$, where $C$ is the constant of integration. For example, $\int xdx=\frac{x^2}{2}+C$.

4. Simplification: After applying the rules of integration, the resulting expression may often be simplified by combining like terms or reducing fractions.

5. Constant of Integration: Since the derivative of a constant is zero, when finding the antiderivative, an arbitrary constant $C$ is added to represent any possible constant that was lost during differentiation.

Understanding these concepts allows one to find the antiderivative of a given function systematically.