Find the Antiderivative f(x)=6-18x
The given problem is asking to determine the antiderivative, also known as the indefinite integral, of the function f(x) = 6 - 18x. Essentially, it is asking for a function whose derivative is f(x). The task involves finding a new function F(x) such that the derivative of F(x) with respect to x equals the given function f(x). This process is the reverse of differentiation, and it seeks to identify the original function before it was differentiated to get f(x).
$f \left(\right. x \left.\right) = 6 - 18 x$
Answer
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.
$$F(x) = \int (6 - 18x) \, dx$$
Step 3:
Decompose the integral into simpler parts.
$$\int 6 \, dx - \int 18x \, dx$$
Step 4:
Utilize the constant multiple rule for integration.
$$6 \int dx - 18 \int x \, dx$$
Step 5:
Extract the constant $-18$ outside the integral.
$$6x + C_1 - 18 \int x \, dx$$
Step 6:
Apply the power rule for integration to find the integral of $x$.
$$6x + C_1 - 18 \left( \frac{x^2}{2} + C_2 \right)$$
Step 7:
Proceed to simplify the expression.
Step 7.1:
Combine like terms.
$$6x - 18 \left( \frac{x^2}{2} \right) + C$$
Step 7.2:
Further simplify the expression.
Step 7.2.1:
Multiply $-18$ by $\frac{1}{2}$.
$$6x - \frac{18}{2} x^2 + C$$
Step 7.2.2:
Reduce the fraction by canceling common factors.
Step 7.2.2.1:
Factor out the 2 from $-18$.
$$6x - \frac{2 \cdot (-9)}{2} x^2 + C$$
Step 7.2.2.2:
Eliminate the common factors.
Step 7.2.2.2.1:
Factor out the 2 from the denominator.
$$6x - \frac{2 \cdot (-9)}{2 \cdot 1} x^2 + C$$
Step 7.2.2.2.2:
Cancel out the common factor of 2.
$$6x - \frac{\cancel{2} \cdot (-9)}{\cancel{2} \cdot 1} x^2 + C$$
Step 7.2.2.2.3:
Rewrite the simplified expression.
$$6x - \frac{-9}{1} x^2 + C$$
Step 7.2.2.2.4:
Divide $-9$ by $1$.
$$6x - 9x^2 + C$$
Step 8:
Conclude with the antiderivative of the function $f(x) = 6 - 18x$.
$$F(x) = 6x - 9x^2 + C$$
Knowledge Notes:
Indefinite Integral: The process of finding an antiderivative is known as indefinite integration. The symbol $\int$ denotes integration, and the function being integrated is called the integrand.
Constant Rule: When integrating a constant $a$, the result is $ax + C$, where $C$ is the constant of integration.
Power Rule: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$.
Constant Multiple Rule: If $a$ is a constant and $f(x)$ is a function, then $\int a f(x) \, dx = a \int f(x) \, dx$.
Simplification: This involves algebraic manipulation such as distributing multiplication over addition, combining like terms, and reducing fractions.
Antiderivative: The antiderivative of a function $f(x)$ is another function $F(x)$ whose derivative is $f(x)$. It represents the most general form of the integral, including an arbitrary constant $C$ since differentiation of a constant is zero.
