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.

$\int x^2 \, dx - \int 3x \, dx + \int 2 \, dx$

Step 4:

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

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

Step 5:

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

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

Step 6:

Integrate $x$ using the Power Rule.

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

Step 7:

Apply the constant rule to integrate 2.

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

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 $n \neq -1$, 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 $f(x) = x^2 - 3x + 2$.