Find the Antiderivative f(x)=x^2-5x+6

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.

$$F(x)=\int (x^2-5x+6)dx$$

Step 3:

Decompose the integral into separate integrals for each term.

$$\int x^{2}dx-\int 5xdx+\int 6dx$$

Step 4:

Apply the Power Rule for integration to $x^2$.

$$\frac{x^3}{3}+C_1-\int 5xdx+\int 6dx$$

Step 5:

Factor out the constant $-5$ from the integral.

$$\frac{x^3}{3}+C_1-5\int xdx+\int 6dx$$

Step 6:

Again, use the Power Rule for the integral of $x$.

$$\frac{x^3}{3}+C_1-5(\frac{x^2}{2}+C_2)+\int 6dx$$

Step 7:

Integrate the constant $6$ with respect to $x$.

$$\frac{x^3}{3}+C_1-5(\frac{x^2}{2}+C_2)+6x+C_3$$

Step 8:

Simplify the expression.

Step 8.1:

Combine like terms and constants.

$$\frac{x^3}{3}-\frac{5x^2}{2}+6x+C$$

Step 8.2:

The simplified antiderivative is:

$$F(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+C$$

Step 9:

Arrange the terms in descending order of power.

$$F(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+C$$

Step 10:

Conclude with the final antiderivative of $f(x)=x^2-5x+6$.

$$F(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+C$$

Knowledge Notes:

1. Indefinite Integrals: The antiderivative or indefinite integral is the reverse process of differentiation. It is represented by the integral sign $\int$ followed by the function and the differential $dx$.

2. Power Rule for Integration: This rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, where $n$ is any real number except $-1$.

3. Constant Multiple Rule: This rule allows us to take constants outside the integral. If $k$ is a constant and $f(x)$ is a function, then $\int kf(x)dx=k\int f(x)dx$.

4. Integral of a Constant: The integral of a constant $a$ with respect to $x$ is $ax+C$, where $C$ is the constant of integration.

5. Simplification: After integrating each term, we combine like terms and constants to simplify the expression to its simplest form.

6. Constant of Integration: When finding an indefinite integral, we add a constant term $C$ because the derivative of a constant is zero, and thus the original function could have had any constant added to it.