Question

What is a non elementary integral?

Answer 1

Hint: Here, first we will understand the meaning of elementary functions. We will take some examples of the elementary functions like constant functions, exponential functions, logarithms, trigonometric and inverse trigonometric functions etc. Now, we will understand the term non elementary integrals and see the examples of functions whose integrals are not elementary functions.

Complete step by step solution:
Here we have been asked to describe the term non elementary integrals. First we need to understand the term elementary functions.
Now, in mathematics an elementary function is a function of a single variable composed of particular simple functions. They can also be defined as a sum, product, composition of finitely many polynomials, rational, trigonometric and exponential functions with their inverse function. Examples of elementary functions can be given by the list below: -
(1) Constant functions: - $1,\pi ,e$ etc.
(2) Powers, roots, exponential functions: - $x^2,\sqrt[5]{x},e^x$ etc.
(3) Logarithms: - $\log x$ etc.
(4) Trigonometric, Inverse trigonometric functions: - $\sin x,\cos x,{\sin }^{-1}x,{\cos }^{-1}x$ etc.
(5) Hyperbolic and Inverse hyperbolic functions: - $\sinh x,\cosh x,{\sinh }^{-1}x,{\cosh }^{-1}x$ etc.
(6) The functions obtained by the sum, difference, product and division of the above functions and also forming composite functions using the above functions. For example: - $\sin x+\log x,\cos \left(e^x\right)$ etc.
Now, the functions which are not elementary are called non elementary functions. For example: - error function $erf\left(x\right),{\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^x{e}^{-t^2}dt$, gamma functions $\mathrm{\Gamma }\left(x\right)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{x-1}dt$. The non elementary integral of an elementary function is the anti – derivative of that function which is itself not an elementary function. Listed below are the examples of some functions whose integrals are non elementary: -
(1) Elliptic Integral: - $\sqrt{1-x^4}$.
(2) Logarithmic integral, error function: - ${\displaystyle \frac{1}{\ln x}},e^{-x^2}$.
(3) Frensel, Dirichlet integrals : - $\sin \left(x^2\right),{\displaystyle \frac{\sin x}{x}}$.

Note: Note that we have to study some examples of non elementary integrals and the process to solve them in higher engineering mathematics. Functions like gamma functions, elliptic integrals, error functions, dirichlet integrals etc. You must remember the definitions of all the terms used in the above solution. Some non elementary integral functions are given the name special functions.