Is every elementary function integrable?

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all n-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions. A first simple class of such elementary functions are the polynomials.

Why does X X have no integral?

The number whose square is 2 cannot be expressed in decimal or fractional form using a finite expression. This number is (perhaps) not important enough to be given a name. As Cesareo has said, if the integral of xx had many applications, mathematicians would adopt a name for it.

Is there an unsolvable integral?

There are many that are known to be “unsolvable” (by which I assume you mean can’t be written in terms of other “well-known” functions). The most famous of these is ex2 (or e-x2 ). Similar for sin(x2 ) and cos(x2 ). Another famous example is sin(x)/x.

Which functions Cannot integrate?

Some functions, such as sin(x2) , have antiderivatives that don’t have simple formulas involving a finite number of functions you are used to from precalculus (they do have antiderivatives, just no simple formulas for them). Their antiderivatives are not “elementary”.

What is not an elementary function?

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 are elementary and non elementary functions?

What are elementary integrals?

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.

Why does every elementary function have an elementary derivative?

The short answer is that we have differentiation rules for all the elementary functions, and we have differentiation rules for every way we can combine elementary functions (addition, multiplication, composition), where the derivative of a combination of two functions may be expressed using the functions, their …

Why are some integrals unsolvable?

You’re collecting information from the entire integration domain into a single value. Some functions simply aren’t nice enough to be integrable, other times it’s hard because we impose our desire to represent the result with finitely many letters from a very limited set of symbols.

What does it mean when an integrand has no elementary anti-derivative?

This is what we mean when we say an integrand has no elementary anti-derivative, it means the integral cannot be evaluated using elementary functions, we cannot evaluate the integral and write it in terms of normal functions. Instead we simply say that if one integrates this, then this is what one will obtain.

Does f(x) have no closed form integral?

In the past, I’ve come across statements along the lines of “function f ( x) has no closed form integral”, which I assume means that there is no combination of the operations: which when differentiated gives the function f ( x).

Why is the derivative of a function not easy to integrate?

Because it can’t be integrated in terms of elementary functions. Most functions are ‘not easy’ to integrate in this way. Or, to say the same thing that Dextercioby and Zurtex said, in different words, because there is no elementary function whose derivative is x x!

How to get the anti derivative of x^x?

You won’t get any, since an anti-derivative of x^x is inexpressible in terms of elementary functions. when i plugged it into the integrator of mathematica it gave it back as same…i dont know why it did not do computation.

https://www.youtube.com/watch?v=2cd3nMBuKuM