Problem 40 - Applied Calculus

Question

Find the indefinite integrals. $$ \int 5 e^{z} d z $$

Step-by-Step Solution

Answer

The indefinite integral is \( 5e^{z} + C \).

Step 1: Identify the Function Type

The function inside the integral is an exponential function, specifically of the form \( e^{z} \).

Step 2: Determine the Integration Formula

The integral of an exponential function \( e^{ax} \) with respect to \( x \) is \( \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration. For our problem, \( a = 1 \).

Step 3: Apply the Constant Rule

Before integrating, recognize that the constant 5 can be factored out of the integral. Thus, we have \( \int 5 e^{z} \, dz = 5 \int e^{z} \, dz \).

Step 4: Perform the Integration

Integrate \( e^{z} \) with respect to \( z \). The integral of \( e^{z} \) is itself, thus \( \int e^{z} \, dz = e^{z} + C \).

Step 5: Multiply the Result by the Factor

Multiply the result of the integration by 5, as per the constant rule: \( 5(e^{z} + C) = 5e^{z} + 5C \). However, \( 5C \) is just a constant, so we write it as \( C \).

Step 6: Write the Final Answer

Thus, the indefinite integral is \( 5e^{z} + C \), where \( C \) is the constant of integration.

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form \( e^{z} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions are unique because they have the same rate of change as their value. This property makes them very useful in modeling growth and decay processes.
For example, in science, they model population growth, radioactive decay, and more. The key characteristic of the exponential function is how it rapidly escalates or diminishes as the variable \( z \) increases or decreases.
In the problem above, the exponential function in question is \( e^{z} \). It plays a central role in the integration process, allowing us to utilize specific integration formulas that cater to exponential functions.

Integration Formula

The integration formula for exponential functions is straightforward yet essential. For an exponential function of the form \( e^{ax} \), the indefinite integral with respect to \( x \) is given by:

• \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \)

where \( C \) is the constant of integration, discussed further later.
This formula is important because it tells us how to handle exponential functions when integrating. Notably, in cases where the exponent is just \( z \), or equivalently \( a = 1 \), the formula simplifies to \( \int e^{z} \, dz = e^{z} + C \). Understanding this step not only helps in solving the problem at hand but also equips you to handle various exponential integrals you may encounter.

Constant of Integration

The constant of integration, denoted by \( C \), is crucial in indefinite integrals because they do not have specific limits. When you find an indefinite integral, you're essentially finding a family of functions. These functions are all possible antiderivatives of the integrand.

• The role of \( C \) is to represent any constant that could be added to the antiderivative.
• This means while solving the integral of \( 5e^{z} \), we add \( C \) to account for all vertical shifts in the function's graph.

Without \( C \), the solution would only represent one instance of the many potential solutions. Incorporating \( C \) ensures that our answer remains general and complete. In practical applications, specifying \( C \) often depends on boundary conditions or additional information about the problem's context.