Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{2} e^{x} d x \quad \text { Integration by parts } $$

Step-by-Step Solution

Answer

The integral of \(x^{2} e^{x}\) dx is \(x^{2} e^{x} - 2x e^{x} + 2e^{x} + C\).

Step 1: Apply the Integration by Parts Method

The formula for integration by parts is \(\int u dv = uv - \int v du\). Choose \(u = x^{2}\) and \(dv = e^{x} dx\). Then calculate \(du\) and \(v\). \(du = 2x dx\) and \(v = e^{x}\).

Step 2: Substitute into the Formula

Now, substitute \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula. This results in \(\int x^{2} e^{x} dx = x^{2} e^{x} - \int e^{x} * 2x dx \). The integral on the right side can further be solved with integration by parts.

Step 3: Repeat Integration by Parts

For the remaining integral, choose \(u = 2x\) and \(dv = e^{x} dx\). Then calculate \(du\) and \(v\). \(du = 2 dx\) and \(v = e^{x}\). Substituting into the integration by parts formula yields \(\int 2x e^{x} dx = 2x e^{x} - \int 2e^{x} dx\). Now the formula simplifies to \(x^{2} e^{x} - 2x e^{x} + 2e^{x} + C\), where \(C\) is the constant of integration.

Key Concepts

Indefinite Integral

An indefinite integral, often represented as \( \int f(x)dx \), is a mathematical expression that represents the antiderivative or the original function before differentiation. In simpler terms, it is the reverse process of finding the derivative. Calculating an indefinite integral gives us a family of functions that all have the same derivative, plus a constant term \( C \), known as the constant of integration. This constant accounts for any vertical shifts the original function could have.

Importantly, finding an indefinite integral is like solving a puzzle where we need to figure out the function whose derivative gives us the integrand, the function we wish to integrate. For example, if you're given \( \int x^2 dx \), you're seeking a function whose derivative with respect to \( x \) is \( x^2 \). The solution to this particular integration is \( \frac{x^3}{3} + C \), as the derivative of \( \frac{x^3}{3} \) with respect to \( x \) is indeed \( x^2 \).

Integration Techniques

There are several integration techniques available for computing integrals, with each method suitable for different types of functions or expressions. The method of integration by parts is particularly helpful when the integrand is a product of two functions, where one function is easily integrable, and the other is easily differentiable.

Integration by Parts

The technique of integration by parts is based on the product rule for differentiation and is expressed as \( \int u dv = uv - \int v du \). To effectively use this method, we strategically choose \(u\) and \(dv\) such that the resulting integral \( \int v du \) is simpler than the original. This process may need to be repeated if the new integral still involves a product of functions.

Choosing \(u\) and \(dv\) wisely is key as it can greatly simplify the calculation, and sometimes, a single problem may have multiple valid approaches. The aim is to reduce the integral into simpler parts that are more manageable, and may potentially involve other techniques like substitution or partial fractions.

For the integral \( \int x^2e^x dx \), as given in the original exercise, the application of integration by parts makes the problem solvable step by step.

Exponential Functions

Exponential functions are functions in the form of \( f(x) = a^{x} \), where \( a \) is a positive constant base and \( x \) is the exponent or power. Exponential functions describe a wide range of phenomena in real life, such as compound interest, population growth, and radioactive decay.

One of the most important exponential functions in mathematics is the natural exponential function, \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. The significance of \( e^x \) lies in the fact that it is its own derivative, which makes it a unique and particularly powerful function in calculus.

When integrating exponential functions, especially those with the base \( e \) and multiplied by a polynomial, such as \( x^2e^x \) from our exercise, integration by parts is commonly used because the derivative of \( e^x \) is also \( e^x \) and thus remains manageable after each application of the technique. This integral showcases the elegant relationship between polynomial and exponential functions through the process of integration.