Find the Antiderivative f(x)=x^2-3x+2

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$F(x)=\int f(x)dx$

Step 2:

Write out the integral that needs to be solved.

$F(x)=\int (x^2-3x+2)dx$

Step 3:

Decompose the integral into separate integrals for each term.

$\int x^{2}dx-\int 3xdx+\int 2dx$

Step 4:

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

$\frac{x^3}{3}+C_1-\int 3xdx+\int 2dx$

Step 5:

Factor out the constant $-3$ from the integral.

$\frac{x^3}{3}+C_1-3\int xdx+\int 2dx$

Step 6:

Integrate $x$ using the Power Rule.

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

Step 7:

Apply the constant rule to integrate 2.

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

Step 8:

Simplify the expression.

Step 8.1:

Combine like terms.

$\frac{x^3}{3}+C_1-\frac{3x^2}{2}-3C_2+2x+C_3$

Step 8.2:

Simplify further.

$\frac{x^3}{3}-\frac{3x^2}{2}+2x+C$ (where $C=C_1-3C_2+C_3$)

Step 9:

Reorder the terms for the final antiderivative.

$\frac{1}{3}{x}^3-\frac{3}{2}{x}^2+2x+C$

Step 10:

Conclude with the antiderivative of $f(x)=x^2-3x+2$.

$F(x)=\frac{1}{3}{x}^3-\frac{3}{2}{x}^2+2x+C$

Knowledge Notes:

The process of finding the antiderivative involves several key concepts in calculus:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is found by calculating the indefinite integral, denoted as $\int f(x)dx$. This represents the family of all functions whose derivative is $f(x)$.

2. Power Rule for Integration: To integrate a power of $x$, use the formula $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\ne -1$, where $C$ is the constant of integration.

3. Linearity of Integration: The integral of a sum or difference of functions is the sum or difference of their integrals, i.e., $\int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx$.

4. Constant Multiple Rule: A constant factor can be pulled out of the integral, $\int k\cdot f(x)dx=k\cdot \int f(x)dx$, where $k$ is a constant.

5. Constant Rule: The integral of a constant $k$ is $kx+C$, where $C$ is the constant of integration.

6. Combining Constants: When multiple constants of integration appear in a calculation, they can be combined into a single constant because the antiderivative is defined up to an arbitrary constant.

By applying these principles, one can systematically find the antiderivative of a polynomial function like $f(x)=x^2-3x+2$.