Question

In Exercises \(21-24,\) use tabular integration to find the antiderivative. $$\int\left(x^{2}-5 x\right) e^{x} d x$$

Short answer

\((x^2 - 5x)e^{x} - (2x - 5)e^{x} + 2e^{x} + C\

Step-by-step solution

Differentiate and Integrate

Differentiate \(u = x^2 - 5x\) until you get zero: first derivative is \(2x - 5\), second derivative is \(2\), and the third is \(0\). Integrate \(v = e^{x}\) as many times as you differentiate \(u\): first integral is \(e^{x}\), second integral is \(e^{x}\), and third is \(e^{x}\).

Apply Tabular Integration

Set up the antiderivative table as follows:\n \(u\) | \(v\) \n \(x^2 - 5x\) | \(e^{x}\) \n \(2x - 5\) | \(e^{x}\) \n \(2\) | \(e^{x}\) \n \(0\) | \(e^{x}\)

Calculate the integral

Follow the tabular integration rule of alternate plus and minus: integral = \((x^2 - 5x)e^{x} - (2x - 5)e^{x} + 2e^{x}\)

Add the integration constant

Finally, don't forget to add the integration constant C, where the final integral is \((x^2 - 5x)e^{x} - (2x - 5)e^{x} + 2e^{x} + C\).

Key concepts

Antiderivative

Understanding the concept of an antiderivative is essential when studying calculus. It refers to the reverse process of differentiation, which means finding a function whose derivative is the given function. This process is also known as integration.

For example, if we're given a function, such as a polynomial like \(x^2-5x\), the antiderivative would be a new function whose derivative would return \(x^2-5x\). By knowing the rules of differentiation, we can reverse engineer the process to determine the antiderivative.

Finding the antiderivative is relatively straightforward for simple functions, but it becomes trickier with more complicated expressions, especially when they involve products of functions like polynomials and exponential functions. This is where special methods, such as tabular integration, come into play to simplify the process.

Integration by Parts

Integration by parts is a technique that transforms the integral of a product of functions into an integral that is hopefully easier to evaluate. It is based on the product rule for differentiation and can be written as:
\[\int u dv = uv - \int v du\]
Where \(u\) and \(dv\) are parts of the original integral. The basic strategy is to choose \(u\) and \(dv\) such that the derivative \(du\) and the integral \(v\) are simpler than the original functions. Tabular integration is a streamlined version of this technique, particularly effective when one of the functions can be repeatedly differentiated until it reduces to zero.

In the given exercise, the polynomial \(x^2-5x\) is differentiated, while the exponential function \(e^x\) is integrated. The tabular method allows for an organized approach to applying integration by parts, especially when dealing with higher-order polynomials or multiple applications of the technique.

Exponential Functions

Exponential functions are among the most important in mathematics, characterized by the constant base raised to a variable exponent. The common form is \(f(x) = a^x\), where \(a\) is a constant. If \(a\) is the mathematical constant \(e\) (approximately 2.71828), then the function \(e^x\) represents the natural exponential function, which arises frequently in calculus.

Exponential functions have unique properties, making them particularly interesting. They are their own derivatives and antiderivatives, meaning that the integral and derivative of \(e^x\) is \(e^x\) itself, with some constant factors potentially added. This feature makes them relatively simple to integrate or differentiate.

In the context of integration by parts, when an exponential function is one of the components, as in the exercise with \(e^x\), it is often strategically chosen as the part to be integrated, since integrating \(e^x\) will always result in another \(e^x\), making the tabular method a straightforward way of simplifying the integration process.