Find the Antiderivative f(x)=x^2-9x+4

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.

$\int x^{2}dx-\int 9xdx+\int 4dx$

Step 4:

Apply the Power Rule to integrate $x^2$.

$\frac{x^3}{3}+C-\int 9xdx+\int 4dx$

Step 5:

Factor out the constant $-9$ from the integral.

$\frac{x^3}{3}+C-9\int xdx+\int 4dx$

Step 6:

Integrate $x$ using the Power Rule.

$\frac{x^3}{3}+C-9(\frac{x^2}{2}+C)+\int 4dx$

Step 7:

Apply the constant rule to integrate 4.

$\frac{x^3}{3}+C-9(\frac{x^2}{2}+C)+4x+C$

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:

1. 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.

2. Power Rule for Integration: This rule states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\ne -1$. It is used to integrate polynomial functions term by term.

3. Constant Multiple Rule: This rule allows us to take constants outside the integral, making it easier to integrate functions with constant coefficients. For example, $\int a\cdot f(x)dx=a\cdot \int f(x)dx$.

4. Sum Rule for Integration: This rule allows us to integrate each term of a sum (or difference) separately. For example, $\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$.

5. Constant Rule: The integral of a constant $a$ with respect to $x$ is $ax+C$.

6. Combining Like Terms: After integrating, similar terms are combined and constants are added together to simplify the antiderivative.

7. 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.