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

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.

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

Step 3:

Decompose the integral into simpler integrals.

$$ \int 2x \, dx + \int (-3) \, dx $$

Step 4:

Extract the constant multiplier from the integral of $x$.

$$ 2 \int x \, dx + \int (-3) \, dx $$

Step 5:

Apply the power rule for integration to $x$.

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

Step 6:

Integrate the constant term.

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

Step 7:

Simplify the expression.

Step 7.1:

Combine like terms.

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

Step 7.2:

Final simplification.

$$ x^2 - 3x + C $$

Step 8:

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

$$ F(x) = x^2 - 3x + C $$

Knowledge Notes:

To solve for the antiderivative (also known as the indefinite integral) of a function, we apply the fundamental theorem of calculus and integration techniques. Here are the relevant knowledge points and detailed explanations:

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. Linearity of Integration: The integral of a sum of functions is the sum of their integrals. This allows us to split the integral of $2x - 3$ into the sum of integrals of $2x$ and $-3$.

3. Constant Multiple Rule: If a constant $k$ is multiplied by a function, the integral of the product is $k$ times the integral of the function. This is why we can factor out the $2$ from the integral of $2x$.

4. Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$. In this case, integrating $x$ gives us $\frac{x^2}{2} + C$.

5. Integrating Constants: The integral of a constant $k$ with respect to $x$ is $kx + C$. This is applied to the integral of $-3$ to get $-3x + C$.

6. Simplification: Combining like terms and simplifying expressions are standard algebraic techniques used to express the antiderivative in its simplest form.

By following these steps and applying these rules, we find the antiderivative of the given function $f(x) = 2x - 3$ to be $F(x) = x^2 - 3x + C$.