Problem 47 - Essential Calculus

Question

State whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v\). Explain your reasoning. $$ \int x^{2} e^{2 x} d x $$

Step-by-Step Solution

Answer

Yes, the integral \(\int x^{2} e^{2x} dx\) can be evaluated using integration by parts. For the first application of integration by parts, one would use \(u = x^2\) and \(dv = e^{2x} dx\). The integral can be simplified to \(0.5x^2e^{2x} - \int x e^{2x} dx\). The process needs to be repeated for the resulting integral \(\int x e^{2x} dx\), by taking \(u = x\) and \(dv = e^{2x} dx\).

Step 1: Identify u and dv

Choose \(u\) and \(dv\) for integration by parts. A rule of thumb is to choose \(u\) as a function that becomes simpler when differentiated, and \(dv\) as a function that doesn't become more complicated when integrated. So, it's suggested here to take \(u = x^2\) and \(dv = e^{2x} dx\).

Step 2: Find du and v

After identifying \(u\) and \(dv\), compute \(du\) and \(v\). Here, the differential of \(u\), \( du = 2x dx\), is derived by differentiating \(u\). The function \(v\), \(v = 0.5e^{2x}\), is obtained by integrating \(dv\).

Step 3: Apply the integration by parts formula

After identifying \(u\), \(v\), \(du\), and \(dv\), input them into the formula \(\int u dv = uv - \int v du\). Substituting the known values, we get \(\int x^{2} e^{2x} dx = x^2 * (0.5e^{2x}) - \int 0.5e^{2x} * 2x dx\). This simplifies to \(0.5x^2e^{2x} - \int x e^{2x} dx\).

Step 4: Repeat the process

Notice that the new integral \(\int x e^{2x} dx\) is still a product of a polynomial in \(x\) and an exponential function, so it can again be handled with the method of integration by parts. Again, let \(u = x\), \(dv = e^{2x} dx\), calculate \(du = dx\) and \(v = 0.5e^{2x}\), and apply the formula for integration by parts. The final answer will be reached after this step.

Key Concepts

Integration Techniques

To solve integrals, multiple methods are available. One common approach is integration by parts. This technique is particularly useful when dealing with the product of different types of functions, such as polynomials multiplied by exponential functions. The integration by parts technique is based on the product rule for differentiation. The formula is: \[ \int u \, dv = uv - \int v \, du \]Here is how to use integration by parts for solving a complex integral:

• Identify two parts of the integrand to use as \(u\) and \(dv\).
• Differentiation of \(u\) gives \(du\), while integration of \(dv\) provides \(v\).
• Apply these to the integration by parts formula to simplify and solve the integral.

In the exercise, we had \(\int x^2 e^{2x} dx\). We chose \(u = x^2\) and \(dv = e^{2x} dx\) based on these criteria.' Integration by parts simplifies the solution, transforming the integral into an easier-to-evaluate form and sometimes reducing it to a basic formula if repeated as demonstrated.

Indefinite Integrals

The indefinite integral represents the anti-derivative of a function. When solving indefinite integrals, you will find a family of functions that differentiate to give the original integrand, plus an arbitrary constant, \(C\), since differentiation of constants is zero.
Here's the general representation:\[ \int f(x) \, dx = F(x) + C \]Where \(F(x)\) is any function whose derivative is \(f(x)\). Finding indefinite integrals often involves methods like substitution or integration by parts. It requires persistence as there could be rounds of transformations to solve successfully, especially if products of functions are involved.

In our example, starting with \(\int x^2 e^{2x} \, dx\), the indefinite nature enables working through repeated integration by parts to eventually reach a solution. The choice of \(u\) and \(dv\) helps guide the strategy and eventually leads to an expression containing the original form of the integrand reduced to simpler operations.

Exponential Functions

Exponential functions are key players when performing integration, especially in integration by parts, owing to their unique derivative and integral properties. A fundamental form of an exponential function is \( e^{ax} \), where "\(e\)" is Euler's number, a constant approximately equal to 2.718. These functions have a wonderful property: they remain identical in form when you differentiate or integrate them. This means:\[ \frac{d}{dx} e^{ax} = ae^{ax} \]and \[ \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \]In our scenario with \( e^{2x} \), recognizing and utilizing these properties simplifies the problem substantially. Applying the rules correctly, you can integrate the function without complications. This simplification is harnessed while using integration by parts, providing a seamless balance: one part gets simpler while the other maintains exponential integrity.