Problem 321 - Calculus Volume 1

Question

In the following exercises, compute each indefinite integral. $$\int e^{-3 x} d x$$

Step-by-Step Solution

Answer

\(-\frac{1}{3}e^{-3x} + C\)

Step 1: Identify the Integral Form

The given integral is \(\int e^{-3x} \, dx\). This is an integral of the form \(\int e^{ax} \, dx\), where \(a\) is a constant. We need to apply the basic integration rule for exponential functions.

Step 2: Apply the Integral Rule for Exponential Functions

The integral of \(e^{ax}\) is \(\frac{1}{a}e^{ax}\), where \(a\) is a constant. In this problem, \(a = -3\). Thus, the integral becomes \(\int e^{-3x} \, dx = \frac{1}{-3}e^{-3x}\).

Step 3: Simplify the Result

We simplify the expression from Step 2. The factor \(\frac{1}{-3}\) can be rearranged to \(-\frac{1}{3}\). So the integral becomes \( -\frac{1}{3}e^{-3x} \).

Step 4: Add the Constant of Integration

For indefinite integrals, always add a constant of integration, \(C\), to account for any constant term that differentiates to zero. Thus, the final result is \(-\frac{1}{3}e^{-3x} + C\).

Key Concepts

Exponential Functions

Exponential functions are a class of mathematical functions characterized by their unique growth behavior, where the variable appears as an exponent. In the realm of calculus, they are often denoted as \( e^{ax} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. This base has unique properties that make exponential functions particularly easy to differentiate and integrate compared to other functions.
Exponential functions are widely used in real-world applications, ranging from compound interest calculations in finance to modeling population growth in biology. A key property of \( e^{x} \) is that it is its own derivative, meaning the rate of change of the function is proportional to the function itself.
When considering the indefinite integral of an exponential function, understanding this self-replicating nature helps simplify the integration process. This is especially true for functions of the form \( e^{ax} \), where \( a \) is a constant. Their integration follows a straightforward rule, making problem-solving with these functions more manageable.

Basic Integration Rule

The basic integration rule for exponential functions is crucial for evaluating indefinite integrals involving terms like \( e^{ax} \). When integrating \( e^{ax} \), where \( a \) is a constant, the rule is:

• The integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \),

where \( C \) is the constant of integration that captures the family of antiderivatives.
In this rule:

• The constant \( a \) in the exponent is factored out, creating the \( \frac{1}{a} \) coefficient in the result,
• This coefficient arises because we reverse the chain rule used during differentiation, adjusting the integral to account for the derivative of \( ax \).

Using this basic integration rule allows for direct computation of indefinite integrals of exponential functions, which would otherwise require more complex techniques. Such clarity in integration rules helps students to take on problem-solving more confidently.

Constant of Integration

Whenever you compute an indefinite integral, adding a constant of integration, denoted as \( C \), is crucial. This constant accounts for the fact that antiderivatives are not unique.

• Indefinite integrals represent families of functions that only differ by a constant.
• When differentiating a function, any constant term vanishes since its derivative is zero.

Hence, when reversing this process through integration, we recognize that there could have been an "invisible" constant.
Including the constant of integration ensures that all possible antiderivatives are represented. Without this step, an integral solution would be incomplete and potentially misleading in applications that require precise computations. Remember, this constant is applicable only for indefinite integrals, not definite ones, where the limits of integration eliminate the need for \( C \).