Problem 82 - Calculus and its applications

Question

Solve each integral. Each can be found using rules developed in this section, but some algebra may be required. $$ \int b e^{a x} d x $$

Step-by-Step Solution

Answer

\( \frac{b}{a} e^{a x} + C \)

Step 1: Identify the Integral Type

The integral \[ \int b e^{a x} \, dx \] involves an exponential function. This indicates that we can use integration rules applicable to exponential integrals, particularly the rule for integrating functions of the form \( e^{ax} \).

Step 2: Use the Constant Multiple Rule

According to the constant multiple rule, a constant can be factored out of the integral. Thus, we can rewrite our integral as:\[b \int e^{a x} \, dx\] where \( b \) is a constant.

Step 3: Integrate the Exponential

The integral for \( \int e^{a x} \, dx \) is \( \frac{1}{a} e^{a x} + C \), where \( C \) is the constant of integration. Therefore, integrating yields:\[b \times \frac{1}{a} e^{a x} + C\]

Step 4: Simplify the Expression

Simplify the integrated expression by multiplying the constants:\[\frac{b}{a} e^{a x} + C\] This is the simplified expression of the integral with \( b \) factored in.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent, such as in the function \( e^{ax} \). These functions are incredibly important in calculus due to their unique properties. They grow rapidly and have the distinctive characteristic that the rate of change is directly proportional to their value.
In calculus, exponential functions are often encountered in problems involving growth and decay models, such as population growth or radioactive decay. The base \( e \), also known as Euler's number, is approximately 2.718. It is the most commonly used base for these types of calculations due to its natural occurrence in mathematical models of continuous growth.
When you integrate an exponential function, you aim to determine the antiderivative or a function whose derivative gives the original function back. The integral of \( e^{ax} \) is a foundational idea in calculus, given by \( \frac{1}{a} e^{ax} + C \), where \( C \) represents a constant of integration. This formula reflects how the integral of an exponential function correlates to its derivative.

Integration Techniques

Integrating functions involves finding a function that, when differentiated, yields the original function given. There are several techniques to accomplish this, each suited to specific kinds of functions.
For exponential functions like \( e^{ax} \), the integration process uses a straightforward rule. Key steps in these techniques include recognizing the type of function you are integrating and applying specific formulas or rules that simplify the work. For instance, the integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \).
Another crucial method is substitution, which simplifies integrals by changing variables to make an integral easier to evaluate. However, for basic exponential integrals, direct application of formulas works beautifully. Mastering these techniques not only helps solve integrals rapidly but also reinforces a deeper understanding of mathematical processes involved in calculus.

Constant Multiple Rule

The constant multiple rule in integration is a handy tool that helps streamline the process of integration when constants are involved. If you have an integral such as \( \int b \, f(x) \, dx \), where \( b \) is a constant, the rule allows you to move \( b \) outside of the integral: \( b \int f(x) \, dx \). This simplification is powerful because it breaks down complex problems into manageable parts.
Applying this rule makes calculations easier, as seen in the original exercise where \( b \) was factored out: \( b \int e^{ax} \, dx \). By recognizing and employing the constant multiple rule, you focus on integrating the function itself without worrying about constant coefficients.
This approach not only provides clarity during problem-solving but also highlights the elegance and efficiency that integration operations can offer. Employing this rule ensures that you do not perform unnecessary calculations, leading to more accurate and fast resolutions in calculus.