Table of Contents
- 1 What is a function that Cannot be integrated?
- 2 Is there a function that Cannot be differentiated?
- 3 Can all the functions in mathematics be integrated?
- 4 What functions can be differentiated?
- 5 Can all functions be differentiated?
- 6 Why we integrate any function?
- 7 Can funfunctions be Riemann integrated?
- 8 What are the non-integrable functions of rational numbers?
- 9 What is the difference between non-integrable and Riemann integration?
What is a function that Cannot be integrated?
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”.
Is there a function that Cannot be differentiated?
A function that does not have a differential. The continuous function f(x)=xsin(1/x) if x≠0 and f(0)=0 is not only non-differentiable at x=0, it has neither left nor right (and neither finite nor infinite) derivatives at that point.
Can all the functions in mathematics be integrated?
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.
Why is integration difficult?
The problem is that differentiation of elementary functions always involves elementary functions; however, integration (anti-derivative) of elementary function may not involve elementary functions. This is the reason why the process of integration is, in general, harder.
What functions do not have antiderivatives?
Examples of functions with nonelementary antiderivatives include:
- (elliptic integral)
- (logarithmic integral)
- (error function, Gaussian integral)
- and (Fresnel integral)
- (sine integral, Dirichlet integral)
- (exponential integral)
- (in terms of the exponential integral)
- (in terms of the logarithmic integral)
What functions can be differentiated?
How to apply the rules of differentiation
Type of function | Form of function | Rule |
---|---|---|
y = constant | y = C | dy/dx = 0 |
y = linear function | y = ax + b | dy/dx = a |
y = polynomial of order 2 or higher | y = axn + b | dy/dx = anxn-1 |
y = sums or differences of 2 functions | y = f(x) + g(x) | dy/dx = f'(x) + g'(x). |
Can all functions be differentiated?
In theory, you can differentiate any continuous function using 3. The Derivative from First Principles. The important words there are “continuous” and “function”. You can’t differentiate in places where there are gaps or jumps and it must be a function (just one y-value for each x-value.)
Why we integrate any function?
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.
Why integration is harder than differentiation?
Integration is generally much harder than differentiation. This little demo allows you to enter a function and then ask for the derivative or integral. You can also generate random functions of varying complexity. Integration typically takes much longer, if the process completes at all!
Are all integrable functions analytic functions?
Since analytic functions are very “nice” and the only requirement for integrability is that the function be bounded and have discontinuities only on a set of measure 0, jumping form integrable to analytic leaves out “almost all” integrable functions! What function are you talking about? And what do you mean by “expanding it over infinite terms”?
Can funfunctions be Riemann integrated?
Functions that have discontinuities of positive measure can not be Riemann integrated, for instance the characteristic function of the rational numbers on the unit interval. However, a more general notion of integral, the Lebesque integral, includes this function and many other that can not be Riemann integrated.
What are the non-integrable functions of rational numbers?
These are non-integrable. Functions that have discontinuities of positive measure can not be Riemann integrated, for instance the characteristic function of the rational numbers on the unit interval. However, a more general notion of integral, the Lebesque integral, includes this function and many other that can not be Riemann integrated.
What is the difference between non-integrable and Riemann integration?
Some function’s integrals however do not have well defined limits on unbounded domains. These are non-integrable. Functions that have discontinuities of positive measure can not be Riemann integrated, for instance the characteristic function of the rational numbers on the unit interval.