What does no elementary antiderivative mean?
You may have meant to ask “what kinds of functions DO have antiderivatives but their antiderivatives are not elementary functions?” — meaning that they cannot be expressed using the kinds of functions you have explored in a typical early Calculus class. The question as asked is surprisingly tough to answer completely.
What are elementary and non elementary functions?
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field …
What is an elementary integral?
By an integral-elementary function we mean any real function that can be obtained from the constants, sin x , ex , logx , and arcsin x (defined on (−1,1 )) using the basic algebraic operations, composition and integration.
Which functions do not have antiderivative?
Liouville’s theorem: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is e−x2, whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics.
Do all functions have integrals?
Not every function can be integrated. Some simple functions have anti-derivatives that cannot be expressed using the functions that we usually work with. One common example is ∫ex2dx.
What is non elementary school?
: not elementary especially : not of or relating to an elementary school a nonelementary teacher.
How many types of elementary functions exist?
Seven Elementary Functions and Their Graphs – Concept.
Are all integrals solvable?
For an integral, most of the ones you’ll see are almost always going to be solvable. Indefinite ones are the ones you might be worried about, since there are no safety nets in that case, but with definite integrals there are often various tricks that you may use to find the value of the integral.
Do all continuous functions have integrals?
A lot more functions have integrals than have derivatives. Any function that is bounded (doesn’t go off to infinity in the interval) and continuous, or even continuous with finitely many jumps, has an integral.
Why can’t x x be integrated?
But we don’t have one for a function which has an x in both the base and the power. We can take the derivative of it just fine, but trying to take its integral is impossible because of the lack of rules it would work with. Therefore, you don’t actually get a function to determine it.