Find the Antiderivative f(x)=x/2

Solution:

Step 1:

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

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

Step 2:

Prepare the integral for computation.

$$ F(x) = \int \frac{x}{2} \, dx $$

Step 3:

Extract the constant $\frac{1}{2}$ from the integral as it does not depend on $x$.

$$ \frac{1}{2} \int x \, dx $$

Step 4:

Apply the Power Rule for integration to find the integral of $x$.

$$ \frac{1}{2} \left( \frac{x^2}{2} + C \right) $$

Step 5:

Proceed to simplify the expression.

Step 5.1:

Express $\frac{1}{2} \left( \frac{x^2}{2} + C \right)$ as $\frac{1}{2} \cdot \frac{x^2}{2} + C$.

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

Step 5.2:

Carry out the simplification.

Step 5.2.1:

Multiply $\frac{1}{2}$ by $\frac{x^2}{2}$.

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

Step 5.2.2:

Simplify the fraction.

$$ \frac{1}{4} x^2 + C $$

Step 6:

Conclude with the antiderivative of $f(x) = \frac{x}{2}$.

$$ F(x) = \frac{1}{4} x^2 + C $$

Knowledge Notes:

The process of finding the antiderivative, or indefinite integral, of a function involves reversing the operation of differentiation. When integrating a function, we are essentially looking for a function whose derivative is the given function. The antiderivative is not unique; it includes a constant of integration $C$ because the derivative of a constant is zero.

Here are some relevant knowledge points and explanations:

1. Indefinite Integral: The general form of an antiderivative, represented by $\int f(x) \, dx$, where $f(x)$ is the function to be integrated.

2. Constant Multiple Rule: When integrating a constant multiplied by a function, the constant can be factored out of the integral. For example, $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.

3. 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 the reverse of the power rule for differentiation.

4. Constant of Integration ($C$): Since the derivative of a constant is zero, any constant can be added to the antiderivative. This constant $C$ represents an infinite number of possible antiderivatives.

5. Simplification: After applying the rules of integration, the resulting expression may often be simplified to a more compact form by combining like terms or reducing fractions.

In the given problem, the function $f(x) = \frac{x}{2}$ is integrated by first factoring out the constant $\frac{1}{2}$, then applying the power rule, and finally simplifying the expression to arrive at the antiderivative $F(x) = \frac{1}{4} x^2 + C$.