1. A System for Integration Formulas
Richa Sarin
Senior Project, Math/Computer Science
Advisor: Prof. Robert Mayans

2. Presentation Outline Introduction Integration examples and solutions
Purpose of the project
Integration system, examples
Architecture
Further work
Conclusions

3. Some Integration Problems

4. How do we find the formulas?
Take Calculus I, II, III.
Look them up in a table of integrals.
Use of computer algebra systems.
Maple
Derive
Mathematica

5. How do we find the formulas?
Some examples from Derive:

6. How do we find the formulas?
Sometimes the formulas can be very complicated!
Integral:
Solution:
What does this answer tell you?

7. How do we find the formula?
It is hard to tell the structure of the answer from a complicated formula.
“Integration is the most esoteric part of mathematics.”
The mathematics of how you get the integral is missing!

8. How to find the formulas?
The symbolic answer does not tell you the underlying concepts and patterns
Example: Last integral is a Chebyshev integral
Form:

9. How to find the formulas?
The symbolic answer does not tell you the techniques, such as substitution, that get you the answer.
Example:
Replace sin2(x) and substitute for cos2(x)

10. Aims of the project A different approach: Integral  Form  Text
The integral is matched to a general form.
Example:
The form links to an explanatory text.
The text explains the derivation of the formula and links to other mathematical texts.

11. Aims of the project Example: Integral:
Form: where p,q,r are rational numbers.
∫ Integral  Form  Text

12. Aims of the project
Form
Text:
This integral is called the binomial integral or the Chebyshev integral, and Chebyshev first proved the conditions under which the integral is an elementary function.
First, we substitute u=xq to bring the integral to the form.
Let s=(r+1)/(q-1). Chebyshev proved that this integral is an elementary function iff s, p, or s+p is an integer.

13. Web Architecture
The Mathematics Hypertext Project (MHP) is an interconnected structure of Web pages of mathematical texts.
An interactive HTML page (part of the MHP) reads the integral from the user input.

14. Web Architecture
A CGI script in Perl on a university server processes the integral and produces a new page.
This page contains the list of forms matched by the CGI scripts.
The user follows the links to the explanatory texts in the MHP.

15. CGI Script/ Web Architecture
Server
Client
CGI script
Pattern matcher
New page
MHP Web pages

16. Software Architecture
Integrals  Form  Text
Want to integrate
Each pattern has its own webpage and explanations

17. Software Architecture
Function of the CGI script:
Reads the integrand from the user input
Scans and parses the integrand into an expression tree.
Reads in a file of forms and references to the texts
Matches the expression to the forms
Lists all the matched forms and links to text into the new page.

18. Some Sample Forms Form: ∫P(x)dx , where P is a polynomial
Pattern: [P x]
Matches: ∫x3 dx, ∫a(x-1)1000 dx
Form: ∫cosn(x) sinm(x) dx, where n,m are positive integers
Pattern: [* [^ [cos x] n] [^ [sin x] m]]]
Matches: ∫sin3(x) cos(x) dx

19. Forms
FORM
EXAMPLE

20. Further work Finish the texts Better pattern matching
Example: sqrt(x) is the same as x1/2
Example: can match an xn but not (ax)n
Google problem: too many matches
Example: integral of x matches xp, P(x), R(x), many more.
Explicit steps from the integrand to the answer.

21. Conclusions System works for simple integrals.
Aims for a better understanding of integration.
The subject of indefinite integrals is vast.