Find the Antiderivative f(x)=-12x
The given problem is asking to determine the antiderivative, also known as the indefinite integral, of the function \( f(x) = -12x \). In essence, the task is to find a function \( F(x) \) such that its derivative \( F'(x) \) is equal to \( f(x) = -12x \). The antiderivative encountered will include a constant term \( C \), since the derivative of any constant is zero and thus does not affect the differentiation process.
$f \left(\right. x \left.\right) = - 12 x$
Answer
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.
$$ F(x) = \int -12x \, dx $$
Step 3:
Extract the constant $-12$ from the integral as it does not depend on $x$.
$$ -12 \int x \, dx $$
Step 4:
Apply the Power Rule for integration to find the integral of $x$.
$$ -12 \left( \frac{x^2}{2} + C \right) $$
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 $1$.
$$ -6 x^2 + C $$
Step 6:
Conclude with the antiderivative of the function $f(x) = -12x$.
$$ F(x) = -6 x^2 + C $$
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:
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., $\int f(x) \, dx$.
Constant Factor Rule: When integrating a function multiplied by a constant, the constant can be pulled out of the integral, e.g., $\int cf(x) \, dx = c \int f(x) \, dx$.
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, $\int x \, dx = \frac{x^2}{2} + C$.
Simplification: After applying the rules of integration, the resulting expression may often be simplified by combining like terms or reducing fractions.
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.
