Find the Antiderivative f(x)=x^2-9x+4
The given problem is a calculus question that asks you to calculate the antiderivative (or indefinite integral) of a polynomial function. Specifically, you need to find a function F(x) such that its derivative, dF/dx, is equal to the original polynomial function f(x) = x^2 - 9x + 4. Calculating the antiderivative involves finding the original function whose rate of change (derivative) corresponds to the given function. This process is also known as integration. The question aims to determine the expression for F(x) that satisfies the condition that the derivative of F(x) with respect to x equals the given polynomial f(x).
$f \left(\right. x \left.\right) = x^{2} - 9 x + 4$
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.
$F(x) = \int (x^2 - 9x + 4) \, dx$
Step 3:
Decompose the integral into separate integrals for each term.
$\int x^2 \, dx - \int 9x \, dx + \int 4 \, dx$
Step 4:
Apply the Power Rule to integrate $x^2$.
$\frac{x^3}{3} + C - \int 9x \, dx + \int 4 \, dx$
Step 5:
Factor out the constant $-9$ from the integral.
$\frac{x^3}{3} + C - 9\int x \, dx + \int 4 \, dx$
Step 6:
Integrate $x$ using the Power Rule.
$\frac{x^3}{3} + C - 9\left( \frac{x^2}{2} + C \right) + \int 4 \, dx$
Step 7:
Apply the constant rule to integrate 4.
$\frac{x^3}{3} + C - 9\left( \frac{x^2}{2} + C \right) + 4x + C$
Step 8:
Simplify the expression.
Step 8.1:
Combine the constants and the $x^2$ term.
$\frac{x^3}{3} + C - \frac{9x^2}{2} + 4x + C$
Step 8.2:
Final simplification.
$\frac{x^3}{3} - \frac{9x^2}{2} + 4x + C$
Step 9:
Arrange the terms in descending order of power.
$\frac{x^3}{3} - \frac{9x^2}{2} + 4x + C$
Step 10:
Present the antiderivative of $f(x) = x^2 - 9x + 4$.
$F(x) = \frac{x^3}{3} - \frac{9x^2}{2} + 4x + C$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves several key concepts in calculus:
Indefinite Integral: The antiderivative of a function $f(x)$ is represented by the indefinite integral $\int f(x) \, dx$. It is called "indefinite" because it includes an arbitrary constant $C$, since the derivative of a constant is zero.
Power Rule for Integration: This rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$. It is used to integrate polynomial functions term by term.
Constant Multiple Rule: This rule allows us to take constants outside the integral, making it easier to integrate functions with constant coefficients. For example, $\int a \cdot f(x) \, dx = a \cdot \int f(x) \, dx$.
Sum Rule for Integration: This rule allows us to integrate each term of a sum (or difference) separately. For example, $\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$.
Constant Rule: The integral of a constant $a$ with respect to $x$ is $ax + C$.
Combining Like Terms: After integrating, similar terms are combined and constants are added together to simplify the antiderivative.
Reordering Terms: It is often customary to write the antiderivative in descending order of the power of $x$.
By applying these rules and principles, one can systematically find the antiderivative of a polynomial function.
