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

Solution:

Step 1:

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

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

Step 2:

Write down the integral that needs to be solved.

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

Step 3:

Decompose the integral into simpler parts.

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

Step 4:

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

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

Step 5:

Extract the constant multiplier from the integral.

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

Step 6:

Integrate $x$ using the Power Rule.

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

Step 7:

Simplify the expression.

Step 7.1:

Combine terms.

$$\frac{x^3}{3} - 2\left(\frac{x^2}{2}\right) + C$$

Step 7.2:

Simplify further.

Step 7.2.1:

Combine the constants.

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

Step 7.2.2:

Simplify the coefficients.

Step 7.2.2.1:

Factor out the common factor.

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

Step 7.2.2.2:

Reduce the fraction.

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

Step 7.2.2.3:

Finalize the simplification.

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

Step 8:

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

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

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. Here are the relevant knowledge points:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is represented by the integral sign without upper and lower limits, denoted as $\int f(x) \, dx$. It represents a family of functions, where $C$ is the constant of integration.

2. Power Rule for Integration: To integrate a power of $x$, the formula is $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$. This rule is a direct consequence of the reverse process of differentiation.

3. Linearity of Integration: The integral of a sum or difference of functions is the sum or difference of their integrals. This property allows us to split integrals into simpler parts, as shown in the solution.

4. Constant Multiple Rule: A constant multiplier can be factored out of the integral, which simplifies the integration process. For example, $\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$, where $k$ is a constant.

5. Simplification: After integrating, it's often necessary to simplify the expression by combining like terms and reducing fractions.

6. Constant of Integration: Since the derivative of a constant is zero, when we find the antiderivative, we must add an arbitrary constant $C$ to account for all possible antiderivatives.