Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int e^{4 x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(e^{4x}\) with respect to \(x\) is \(\frac{1}{4}e^{4x} + C\).

Step 1: Identify the form of the function

The given function is in the form of \(e^{ax}\), where \(a = 4\) and \(x\) is the variable of integration.

Step 2: Apply the power rule of integration

The rule states that the integral of \(e^{ax}\) with respect to \(x\) is \(\frac{1}{a}e^{ax} + C\). We apply this rule directly to compute the integral, with \(a = 4\).

Step 3: Solve and simplify

Substituting \(a = 4\) into the formula, we obtain \(\frac{1}{4}e^{4x} + C\). This represents the antiderivative, or indefinite integral, of the original function.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable appears in the exponent. A typical example of such a function is \( e^{ax} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( a \) is a constant. These functions appear frequently in calculus due to their unique properties, such as how they grow rapidly over time.
Exponential functions are particularly interesting because their rate of growth is proportional to their current value. This property makes them ideal for modeling phenomena like population growth and radioactive decay. When learning calculus, understanding exponential functions is crucial, as they often simplify the process of differentiation and integration. The integral of an exponential function is straightforward to compute, which is helpful when solving real-world problems.

Power Rule of Integration

The power rule of integration simplifies finding the integral of exponential functions such as \( e^{ax} \). It states that the integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \), where \( a \) is a constant. This rule helps us find antiderivatives without diving into lengthy calculations.
Using the power rule, we can quickly evaluate indefinite integrals of exponential functions. It's essential for students to recognize the form \( e^{ax} \) and apply this rule correctly during exercises. This skill eliminates the need for more complex integration techniques, saving time and effort.

• Identify \( a \) as the constant multiple in the exponent.
• Use the power rule to write the antiderivative: \( \frac{1}{a}e^{ax} + C \).
• Don't forget to include \( C \), the constant of integration, which accounts for any vertical shift in the function.

Antiderivatives

An antiderivative, often referred to as an indefinite integral, is a function whose derivative is the original function that you started with. Finding an antiderivative reverses the process of differentiation.
For example, if we find the antiderivative of \( e^{4x} \), it means we are searching for a function whose derivative returns \( e^{4x} \). In this case, the function \( \frac{1}{4}e^{4x} + C \) satisfies this condition. Here, \( C \) is called the constant of integration. It ensures that we account for all possible functions shifted vertically, as differentiation would eliminate such constants.
Understanding antiderivatives is fundamental in calculus because they are used to solve problems related to area under a curve and accumulated quantities. They provide the foundational tools for interpreting changes over an interval and connecting derivative functions back to their original forms.