Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(e^{t}+t^{e}\right) d t$$

Step-by-Step Solution

Answer

The indefinite integral of the given function can be expressed as \(\int (e^t + t^e) dt = e^t + I(t) + C\), where \(I(t)\) represents the special function for the integral of \(t^e\) with respect to \(t\), and \(C\) is the constant of integration.

Step 1: Integrate the first term \(e^t\) with respect to \(t

Since the integral of \(e^t\) with respect to \(t\) is simply \(e^t\), we have \[\int e^t \, dt = e^t + C_1\] 2.

Step 2: Integrate the second term \(t^e\) with respect to \(t

Unfortunately, there is no closed-form expression for the indefinite integral of \(t^e\) with respect to \(t\) in terms of elementary functions. However, you will find this term in many references designated as an "elementary" integral that cannot be expressed in terms of a closed-form expression. Therefore, we will denote the indefinite integral of \(t^e\) in terms of the special function \(I(t)\). That is, \[\int t^e \, dt = I(t) + C_2\] 3.

Step 3: Combine the integrals and add the constants

Bring the two integrals together and add the constants. The indefinite integral of the given function is \[\int (e^t + t^e) dt = (e^t + C_1) + (I(t) + C_2) = e^t + I(t) + C\] where \(C = C_1 + C_2\) is the constant of integration.

Key Concepts

Integration

Integration is a fundamental concept in mathematics, particularly in calculus, where it serves as a tool to find the area under a curve, among other applications. The process of integration, often termed 'Antidifferentiation', involves finding a function (the integral) whose derivative is the original function. This is symbolically represented as \[ F(x) = \int f(x) \, dx + C \], where \( F'(x) = f(x) \), and \( C \) is a constant, known as the constant of integration. It represents the infinite family of functions that could be differentiated to get back to the original function \( f(x) \).When integrating, the goal is to identify the function that yields the initial function upon differentiation. The process can be intuitive for simple functions, with the result expressed in a closed-form in terms of elementary functions. Yet, for more complex functions, the integration process can become intricate and sometimes might not result in an elementary function at all, necessitating the introduction of special functions for representation. Understanding this boundary between elementary functions and the domain of special functions is crucial for students to comprehend the landscape of integral calculus.

Elementary Functions

Elementary functions form the backbone of the functions we often encounter in calculus. These functions can generally be defined by a finite combination of algebraic operations, exponentials, logarithms, constants, and solutions to algebraic equations. Functions like \( e^x \), \( x^n \) (where \( n \) is a real number), \( \sin(x) \), and \( \log(x) \) are classic examples of elementary functions.It's important to understand that while many functions can be integrated into an elementary function, there are some functions, such as \( t^e \) in our exercise, that do not have a straightforward antiderivative that can be expressed in terms of elementary functions. Recognizing this is an essential step in the comprehension of integration, as it introduces the need for more sophisticated mathematical concepts like special functions, which can describe the behavior of such non-elementary integrals.The integration of elementary functions is taught through various techniques such as substitution, integration by parts, and partial fractions. Mastery of these techniques equips students with the ability to tackle a broad range of problems within calculus. However, one must not get discouraged when encountering integrals that defy these conventional methods—it's simply an indication that we are delving into more complex mathematical territory.

Special Function

Special functions are categories of functions that have established names and importance due to their frequent appearance in various branches of physics, engineering, and mathematics. They often arise in solving differential equations, series expansions, or integrals that do not succumb to the set of elementary functions.Examples of special functions include the Gamma function (\( \Gamma(x) \)), which generalizes factorial to non-integer values; the Error function (\( \text{erf}(x) \)), significant in statistics and probability; and Bessel functions, which appear in wave propagation and static potentials. These functions are typically outlined in extensive mathematical tables or implemented in computer algebra systems.For the exercise given, \( \int t^e \, dt \), traditional methods don't yield a result expressible by elementary functions, prompting the introduction of \( I(t) \), a placeholder for a special function that we don't have an elementary expression for. While it may seem limiting to not have a concrete form, using special functions like \( I(t) \) allows mathematicians and scientists to continue working with and understanding behaviors of complex systems where direct solutions are elusive. Introducing and familiarizing oneself with these functions is an integral part of higher-level mathematics education, expanding one's toolkit beyond the realm of basic integrals.