Problem 13 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The indefinite integral of \(x e^{4x}\) is \(\frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x} + C\).

Step 1: Identify the Functions

We first identify our two functions in the given integral. Here, \(f(x) = x\) and \(g'(x) = e^{4x}\). These designations stem from the rule of thumb, which states to take \(u\) as the function that will become simpler when differentiated, which is \(x\) in this case.

Step 2: Apply Integration by Parts Formula

We recall the integration by parts formula, which is \(\int u dv = uv - \int v du\), where \(u = f(x)\) and \(dv = g'(x) \, dx\). So now, we need to find \(du\) and \(v\). We have \(du = f'(x) \, dx = dx\) and \(v = \int g'(x) \, dx = \int e^{4x} \, dx = \frac{1}{4} e^{4x}\).

Step 3: Apply these Values to the Formula

By substituting these values into the integration by parts formula, we get \(\int x e^{4x} \, dx = x (\frac{1}{4} e^{4x}) - \int \frac{1}{4} e^{4x} \, dx = \frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x}\).

Step 4: Add the Integration Constant

As the last step, we need to add the integration constant, \(C\), to our integral. The final result should thus be \(\frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x} + C\).

Key Concepts

Integration by Parts

Integration by parts is a powerful tool used to solve integrals that are products of two functions, which cannot be easily integrated otherwise. This technique relies on the product rule for differentiation and, thought of informally, involves integrating one function and differentiating another.

In our example, the integral of the product of a polynomial and an exponential function, \( \int x e^{4x} dx \), was expertly tackled using this method. The integration by parts formula, \( \int u dv = uv - \int v du \) comes into play, where the choice of \(u\) and \(dv\) is crucial for simplifying the process. \(u\) is often selected to be the function that simplifies upon differentiation, which, in our case, is \(x\). After differentiating \(u\) and integrating \(dv\), the formula is applied to find the solution.

Exponential Functions Integration

The integration of exponential functions follows specific rules that make the process straightforward. The general form is \( \int e^{kx} dx = \frac{1}{k} e^{kx} + C \) where \(k\) is a constant.

Exponential functions are unique because they are their own derivative, which greatly simplifies their integration. In our example, integrating \( e^{4x} \) yields \( \frac{1}{4} e^{4x} \) because the derivative of \( 4x \) is \(4\), and we divide by this coefficient to reverse the differentiation. Understanding the integration of exponential functions is crucial for many calculus problems, including those that require integration by parts.

Integration Rules

There are several integral rules that help when solving integration problems. Some of these include linearity of integration, the power rule, and trigonometric integrals.

Knowing when and how to apply these rules is vital in finding the solution to an integral. For instance, knowing that \( \int kf(x) dx = k \int f(x) dx \) where \(k\) is a constant can simplify the process. Our exercise applied this knowledge by factoring out constants to simplify the integral. Mastery of these rules allows one to approach integration with a toolkit of methods to make the process more efficient.

Integration Constant

The integration constant \(C\) is a fundamental concept in indefinite integrals. Whenever we perform an integration, we're finding the antiderivative or a family of functions whose derivative gives the original function.

Because differentiation eliminates any constant (since the derivative of a constant is zero), when we integrate, we must acknowledge the possibility of a constant having been present in the original function. Hence, we add the integration constant \(C\) at the end of our indefinite integral to represent all possible constant values. As seen in the solution of our example, the constant \(C\) is added to ensure that the antiderivative encompasses all possible functions that would differentiate to the given integrand.