data-ad-format="horizontal">



 
Functions defined using integrals

Not all integrals have antiderivatives that can be written down using the basic functions and operations that we know. Some functions are, in fact, defined using an integral. This page shows two such functions.

Substitute image

This device cannot display Java animations. The above is a substitute static image
See About the calculus applets for operating instructions.

1. Sine integral

The applet shows a graph on the left of (sin x)/x and on the right the graph of sine integral This function, defined using an integral, is called the Sine Integral and frequently written as Si(x). There is no antiderivative for the integrand that can be written using the basic operations of algebra and the basic functions we know; the only way to define the sine integral is using this integral. Move the x slider to see various values of the function and the area that is accumulated to generate that value.

2. Error function

Select the second example, showing the error function, erf This is another function that is defined using an integral. Move the x slider to see various values of the function and the area that is accumulated to generate that value.

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref

Other 'Constructing Antiderivatives' topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.