Find the Antiderivative f(z)=5e^z

Solution:

Step 1:

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

$$F(z) = \int f(z) \, dz$$

Step 2:

Write down the integral that needs to be solved.

$$F(z) = \int 5e^z \, dz$$

Step 3:

Extract the constant multiple from the integral.

$$5 \int e^z \, dz$$

Step 4:

Compute the integral of the exponential function.

$$5(e^z + C)$$

Step 5:

Combine terms to express the antiderivative in its simplest form.

$$5e^z + C$$

Step 6:

Conclude with the antiderivative of the original function $f(z) = 5e^z$.

$$F(z) = 5e^z + C$$

Knowledge Notes:

To solve for the antiderivative (also known as the indefinite integral) of a function, we follow a systematic process:

1. Understanding Antiderivatives: An antiderivative of a function $f(z)$ is a function $F(z)$ whose derivative is $f(z)$. In other words, $F'(z) = f(z)$. The process of finding $F(z)$ is called integration.

2. Integral Setup: The antiderivative is represented by the integral symbol $\int$, followed by the function and the differential variable, in this case, $dz$.

3. Constants in Integration: Constants can be factored out of the integral, simplifying the integration process. This is based on the linearity property of integrals.

4. Exponential Function Integration: The integral of $e^z$ with respect to $z$ is $e^z$, since the derivative of $e^z$ is itself. This is a fundamental property of the exponential function.

5. Adding the Constant of Integration: When finding the indefinite integral, we must add a constant term $C$ because the derivative of a constant is zero, and thus any constant could have been present in the original function.

6. Final Expression: The final step is to write the antiderivative in its simplest form, combining the constant multiple with the integrated function and adding the constant of integration.