Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Integration by parts to find the indefinite integral. $$ \int x^{2} e^{2 x} d x $$

Step-by-Step Solution

Answer

\[\frac{1}{2}x^2 e^{2x} - \frac{1}{2}x e^{2x} + \frac{1}{4} e^{2x} + C.\]

Step 1: Identify u and dv and compute corresponding du and v

Let's choose \(u = x^2\), and then \(dv = e^{2x} dx\). When we differentiate \(u\), we get \(du = 2x dx\), and when we integrate \(dv\), we get \(v = \frac{1}{2}e^{2x}\).

Step 2: Apply the integration by parts formula

The formula is \(\int u v dx = u \int v dx - \int (\frac{du}{dx} \int v dx) dx\). Substituting \(u, du, v\) and \(dv\) into the formula yields: \[x^2 * \frac{1}{2}e^{2x} - \int (2x*\frac{1}{2}e^{2x})dx\] which simplifies to \[\frac{1}{2}x^2 e^{2x} - \int xe^{2x} dx.\]

Step 3: Simplify the remaining integral

Notice we are left with a new integral \(\int xe^{2x} dx\), which can again be calculated using integration by parts. If we let \(u = x\) this time and \(dv = e^{2x} dx\) again, then \(du = dx\) and \(v = \frac{1}{2}e^{2x}\). Substituting into the integration by parts formula once more, we will get \[x*\frac{1}{2}e^{2x} - \int (\frac{1}{2}e^{2x})dx.\] The remaining integral simplifies as \[\int (\frac{1}{2}e^{2x})dx = \frac{1}{4}e^{2x}.\] So we end up with \[\frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x}\].

Step 4: Put everything together

Now we combine the results from Step 2 and Step 3. It looks like this: \[\frac{1}{2}x^2 e^{2x} - \left(\frac{1}{2}x e^{2x} - \frac{1}{4} e^{2x}\right). \]It simplifies to \[\frac{1}{2}x^2 e^{2x} - \frac{1}{2}x e^{2x} + \frac{1}{4} e^{2x}\]. Lastly, add the constant of integration \(C\) to account for the indefinite integral which results in \[\frac{1}{2}x^2 e^{2x} - \frac{1}{2}x e^{2x} + \frac{1}{4} e^{2x} + C.\]

Key Concepts

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus. They can be seen as the reverse of differentiation. When we talk about an indefinite integral, we are looking for a function whose derivative gives us the original function under the integral sign. This is often also referred to as finding the antiderivative.

There is no fixed limit to integrate over, hence the term "indefinite." Instead, when you compute the indefinite integral of a function, you add a constant of integration, denoted by "C," to represent an entire family of functions.

For example, if we have a function that gives us \(f(x) = x^2\), the indefinite integral is \(\int x^2 dx = \frac{1}{3}x^3 + C\). The constant C accounts for all possible shifts of the function along the y-axis.

• Integrals can appear simple, like \(\int x dx = \frac{1}{2}x^2 + C\),
• or complex, requiring specific methods to solve them, like integration by parts.

Integration Methods

There are several common methods for solving integrals, each suitable for different types of functions. One of these methods is integration by parts, which is especially useful when dealing with products of functions. The method is based on the product rule for differentiation and follows the formula:

\[\int u \, dv = uv - \int v \, du\]
To apply this technique, it’s important to choose effectively which part of the integral should be \(u\) and which should be \(dv\). Typically, \(u\) is selected to be something that simplifies upon differentiation, and \(dv\) is something that is easily integrated.

In the exercise given:

• We chose \(u = x^2\) and \(dv = e^{2x} dx\).
• Differentiating \(u\), gave us \(du = 2x dx\).
• Integrating \(dv\), resulted in \(v = \frac{1}{2}e^{2x}\).
• Then we followed the formula to find the integral.

Integration by parts can sometimes need to be applied more than once in a given problem, as seen in this exercise when we needed to integrate \(\int xe^{2x} dx\) again.

Exponential Functions

Exponential functions, typically taking the form \(e^{ax}\) where \(a\) is a constant, are a critical element in mathematics. They have unique properties that make them distinct and useful in solving integrals.

One powerful feature of exponential functions is the similarity between the function and its derivative. Specifically, the derivative of \(e^{ax}\) is \(ae^{ax}\). This property is what makes working with exponential functions particularly convenient when integrating.

In calculus, integrating exponential functions like \(e^{2x}\) requires slightly different steps depending on the constants involved. The integral of \(e^{2x}\) is \(\frac{1}{2}e^{2x}\), as seen by investigating the reverse process of differentiation.

• It's important to account for any constant factor that appears before the exponent during integration.
• This property significantly simplifies the integration, provided you keep track of these constants.

Being comfortable with exponential functions and their integration is essential, especially since they frequently appear in various applications, including finance, physics, and population models.