Problem 23 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(21-24,\) use tabular integration to find the antiderivative. $$\int x^{3} e^{-2 x} d x$$

Step-by-Step Solution

Answer

The antiderivative of \(x^3 e^{-2x}\) with respect to \(x\) is \(-\frac{1}{2}x^{3}e^{-2x} - \frac{3}{4}x^{2}e^{-2x} - \frac{3}{4}x e^{-2x} - \frac{3}{8} e^{-2x} + C\).

Step 1: Choose the function to differentiate and the function to integrate

To start, choose the function to be differentiated and the one to be integrated in the tabular integration. In this case, \(x^3\) is selected to differentiate because its derivative will eventually become zero after several differentiations. On the other hand, \(e^{-2x}\) is chosen to integrate because it does not change much when integrated, it's easy to integrate indefinitely.

Step 2: Create the tabular integration table

Next, create a table for the tabular integration. In the left column, start with the function to differentiate at the top and compute its successive derivatives until zero is reached. On the right column, start with the function to integrate and compute its successive integrations. Alternate plus/minus signs beginning with a plus at the top. The table becomes:\n \[ \begin{{array}}{{c c|c}} + & x^3 & e^{-2x} \ - & 3x^2 & -\frac{1}{2}e^{-2x} \ + & 6x & \frac{1}{4}e^{-2x} \ - & 6 & -\frac{1}{8}e^{-2x} \ + & 0 & \frac{1}{16}e^{-2x} \end{{array}} \]

Step 3: Multiply diagonally and sum

For the final step, each term in the left column of the table is multiplied by the term diagonally below it or to its right in the right column. Add or subtract these products according to the signs in the left column. The result is the antiderivative (up to a constant) of the original integrand:\n \[ \int x^3 e^{-2x} dx = x^3 \cdot (-\frac{1}{2}e^{-2x}) - 3x^2 \cdot (\frac{1}{4}e^{-2x}) + 6x \cdot (\frac{-1}{8}e^{-2x}) - 6 \cdot (\frac{1}{16}e^{-2x}) \]

Step 4: Simplify the result

Finally, you simplify the result to give: \[ \int x^3 e^{-2x} dx = -\frac{1}{2}x^{3}e^{-2x} - \frac{3}{4}x^{2}e^{-2x} - \frac{3}{4}x e^{-2x} - \frac{3}{8} e^{-2x} + C \] where \(C\) is the constant of integration.

Key Concepts

Antiderivative

An antiderivative is a function that reverses the process of differentiation. When you're asked to find the antiderivative of a function, you're essentially trying to determine the original function before it was differentiated.
For example, if you know that the derivative of a function is \(x^3 e^{-2x}\), finding the antiderivative means working out what the function was before differentiation.
In practice, this involves methods like tabular integration, making it possible to find antiderivatives for complex functions like the one in our exercise.
When dealing with complicated functions, especially products of polynomials and exponential functions, methods such as tabular integration become essential tools.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative shows how a function changes as its input changes and is a fundamental concept in calculus.
In our example, we start by differentiating \(x^3\) until we reach zero.
This creates a pathway for integration by reducing the degree of the polynomial, which is essential for constructing the tabular integration table.

• Differentiating \(x^3\) gives 3x².
• Continuing, we get 6x, then 6, and finally 0.

Each derivative corresponds to a simpler form of the original function, eventually allowing the integration process to complete efficiently.

Integration

Integration is the reverse process of differentiation. It involves finding the original function given its derivative.
In the context of tabular integration like in our exercise, integration plays a critical role. We choose an exponential function like \(e^{-2x}\), which is easy to integrate repeatedly without changing its form drastically.
The integrations from this function fill one column of our tabular method and provide continuity in the process.

• Integrating \(e^{-2x}\), gives \(-\frac{1}{2}e^{-2x}\).
• Further integration gives \(\frac{1}{4}e^{-2x}\), and so on.

These integrals, combined with the derivatives of the polynomial, form the basis for obtaining the antiderivative of a product like \(x^3 e^{-2x}\).

Indefinite Integrals

Indefinite integrals represent a family of functions and include an arbitrary constant \(C\). This constant arises because differentiation of a constant is zero, so when reversing the process through integration, we must account for any possible constant that might have vanished.
Taking indefinite integrals helps us find all possible antiderivatives of a given function.
In our exercise, the result of the tabular integration process gives us:

• \(-\frac{1}{2}x^{3}e^{-2x}\)
• \(-\frac{3}{4}x^{2}e^{-2x}\)
• \(-\frac{3}{4}x e^{-2x}\)
• \(-\frac{3}{8} e^{-2x}\)

With a constant \(C\), representing the full set of possible antiderivative solutions. This general form is essential when solving integration problems without specific boundary conditions.