Find the Antiderivative f(x)=2x+5

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.

$$ F(x) = \int (2x + 5) \, dx $$

Step 3:

Decompose the integral into simpler parts.

$$ \int 2x \, dx + \int 5 \, dx $$

Step 4:

Extract the constant coefficient from the integral.

$$ 2\int x \, dx + \int 5 \, dx $$

Step 5:

Utilize the Power Rule to integrate $x$.

$$ 2\left( \frac{x^2}{2} + C \right) + \int 5 \, dx $$

Step 6:

Apply the rule for integrating a constant.

$$ 2\left( \frac{x^2}{2} + C \right) + 5x + C $$

Step 7:

Simplify the expression.

Step 7.1:

Combine the constant and the variable term.

$$ 2\left( \frac{x^2}{2} + C \right) + 5x + C $$

Step 7.2:

Final simplification.

$$ x^2 + 5x + C $$

Step 8:

Present the final antiderivative of $f(x) = 2x + 5$.

$$ F(x) = x^2 + 5x + C $$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the operation of differentiation. The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. The general steps for finding the antiderivative include:

1. Integration: The process of finding the antiderivative is called integration. The symbol for integration is $\int$, and the process is the opposite of differentiation.

2. Power Rule for Integration: This rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$, where $C$ is the constant of integration.

3. Constant Multiple Rule: If $k$ is a constant, then $\int k f(x) \, dx = k \int f(x) \, dx$. This allows us to take constants out of the integral.

4. Integral of a Constant: The integral of a constant $a$ with respect to $x$ is $ax + C$, where $C$ is the constant of integration.

5. Simplification: After integrating, it's important to simplify the expression by combining like terms and constants.

6. Constant of Integration: Since the derivative of a constant is zero, when we find the antiderivative, we add an arbitrary constant $C$ to represent any possible constant that could have been present in the original function before differentiation.

In the given problem, the function to integrate is $f(x) = 2x + 5$. The antiderivative is found by applying the power rule for the term $2x$ and the rule for integrating a constant for the term $5$. After integrating each term and simplifying, we obtain the antiderivative $F(x) = x^2 + 5x + C$.