Find the Antiderivative f(x)=6-18x

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.

$$\int 6dx-\int 18xdx$$

Step 4:

Utilize the constant multiple rule for integration.

$$6\int dx-18\int xdx$$

Step 5:

Extract the constant $-18$ outside the integral.

$$6x+C_1-18\int xdx$$

Step 6:

Apply the power rule for integration to find the integral of $x$.

$$6x+C_1-18(\frac{x^2}{2}+C_2)$$

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:

1. 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.

2. Constant Rule: When integrating a constant $a$, the result is $ax+C$, where $C$ is the constant of integration.

3. Power Rule: For any real number $n\ne -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}+C$.

4. Constant Multiple Rule: If $a$ is a constant and $f(x)$ is a function, then $\int af(x)dx=a\int f(x)dx$.

5. Simplification: This involves algebraic manipulation such as distributing multiplication over addition, combining like terms, and reducing fractions.

6. 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.