Examples of Functions with Derivatives but Indefinite Integrals Not Expressible in Known Functions

Many students and mathematicians are curious about functions that have derivatives but whose indefinite integrals cannot be expressed in terms of known elementary functions. This article explores such examples and discusses their fundamental characteristics.

Introduction

When considering real-valued functions defined on an interval, the idea of finding an indefinite integral and a derivative often intertwines. However, certain functions have derivatives that can be found but whose indefinite integrals remain inexpressible in terms of elementary functions. This article delves into these concepts by providing specific examples and clarifying the meaning of these terms.

Understanding Derivatives and Indefinite Integrals

The derivative of a function at a point is the rate of change of the function at that point. In mathematical notation, if f(x) is a function, then its derivative is denoted as f'(x). For a differentiable function, the derivative f'(x) exists for every x in its domain.

The indefinite integral, also known as the antiderivative, of a function f(x) is a function F(x) whose derivative is f(x). In other words, if F'(x) f(x), then F(x) is an antiderivative of f(x). The indefinite integral is generally expressed as:

∫f(x) dx F(x) C, where C is the constant of integration.

Examples of Functions with Derivatives but Indefinite Integrals Not Expressible in Known Functions

A famous example of such a function is exp(x^2). The derivative of exp(x^2) is found using the chain rule:

d/dx (exp(x^2)) 2x exp(x^2).

However, the indefinite integral ∫exp(x^2) dx cannot be expressed in terms of elementary functions. This was proven by Liouville, a French mathematician, and is a well-known result in the field of differential algebra. This function serves as a classic example of a function with an explicit derivative but an indefinite integral that is non-elementary.

Another Example: sin(1/log(x))

Consider the function sin(1/log(x)) for x > 1. This function is differentiable for all x > 1. Its derivative can be found using the chain rule and the quotient rule:

d/dx (sin(1/log(x))) cos(1/log(x)) * (-1/log(x)^2).

However, the indefinite integral ∫sin(1/log(x)) dx cannot be expressed in terms of elementary functions. This is a bit more complex than the previous example but still provides a concrete example of a function with a known derivative but an indeuitable integral.

Further Reflections

It is worth noting that even though these functions cannot be expressed in terms of elementary functions, their derivatives can still be computed and are well-defined. The issue arises from the inability to find a closed-form expression for the antiderivative in terms of known functions.

The existence of such functions highlights the limitations of elementary functions in certain contexts and underscores the importance of having multiple tools and techniques to analyze and understand functions in mathematics.

Conclusion

While many functions have both derivatives and explicitly expressable indefinite integrals, there are also functions whose derivatives exist but whose indefinite integrals cannot be expressed in terms of known functions. Examples like exp(x^2) and sin(1/log(x)) serve as powerful illustrations of this phenomenon. Understanding these functions not only expands our mathematical knowledge but also introduces us to the rich landscape of mathematical analysis and differential algebra.