Find the Antiderivative f(x)=x^2-5x+6
The problem provided is a calculus question asking for the calculation of the antiderivative (also known as the indefinite integral) of the given function f(x) = x^2 - 5x + 6. The task is to find a function F(x) such that the derivative of F(x) with respect to x is equal to the original function f(x). In other words, you are required to determine the function whose rate of change (or slope at any point) corresponds to the quadratic function provided. The process involves reversing the differentiation operation to obtain the original function before it was differentiated.
$f \left(\right. x \left.\right) = x^{2} - 5 x + 6$
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.
$$F(x) = \int (x^2 - 5x + 6) \, dx$$
Step 3:
Decompose the integral into separate integrals for each term.
$$\int x^2 \, dx - \int 5x \, dx + \int 6 \, dx$$
Step 4:
Apply the Power Rule for integration to $x^2$.
$$\frac{x^3}{3} + C_1 - \int 5x \, dx + \int 6 \, dx$$
Step 5:
Factor out the constant $-5$ from the integral.
$$\frac{x^3}{3} + C_1 - 5 \int x \, dx + \int 6 \, dx$$
Step 6:
Again, use the Power Rule for the integral of $x$.
$$\frac{x^3}{3} + C_1 - 5 \left( \frac{x^2}{2} + C_2 \right) + \int 6 \, dx$$
Step 7:
Integrate the constant $6$ with respect to $x$.
$$\frac{x^3}{3} + C_1 - 5 \left( \frac{x^2}{2} + C_2 \right) + 6x + C_3$$
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:
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 $dx$.
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 $-1$.
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 $\int k f(x) \, dx = k \int f(x) \, dx$.
Integral of a Constant: The integral of a constant $a$ with respect to $x$ is $ax + C$, where $C$ is the constant of integration.
Simplification: After integrating each term, we combine like terms and constants to simplify the expression to its simplest form.
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.
