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

A System for Integration Formulas

Richa Sarin

Senior Project, Math/Computer Science

Advisor: Prof. Robert Mayans

Presentation Outline

• Introduction• Integration examples and solutions• Purpose of the project• Integration system, examples• Architecture• Further work• Conclusions

Some Integration Problems

2

13

1

2 2

cos( )

/ 2

Integral Form

, 11

sin ( )1 1

No elementary formula

1erf ( )

2

nn

x

xt

xx dx x dx C n

ndx dx

x Cx x

x dx

e dt x

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

How do we find the formulas?

• Some examples from Derive:4

3

4

xx dx C

1

2

1sin

1dx x C

x

1ln( )dx x C

x

How do we find the formulas?

• Sometimes the formulas can be very complicated!• Integral:• Solution:

• What does this answer tell you?

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!

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:

dxcxbax pqr )(

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)

dxxxdxx )sin())(cos1()(sin 23

3sin ( )x dx

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.

3 2 1 ( )x x dx P x dx

Aims of the project

Example:

• Integral:

• Form: where p,q,r are rationalnumbers.

dxxx )3/2(4)3/1( )1(

∫ Integral Form Text

dxcxbax pqr )(

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 anelementary function iff s, p, or s+p is an integer.

dxcxbax pqr )(

ducubuq

a pq

r

)()1(

)1(

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.

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.

CGI Script/ Web Architecture

cx

x 4

43

Server Client

MHP Web pages

CGI scriptPattern matcher

New page

Software Architecture

• Integrals Form Text

• Want to integrate

• Each pattern has its own webpage and explanations

)(3 xPxx

dxxx )3/2(4)3/1( )1( dxcxbax pqr )( dxxPdxxx )(1242

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.

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

Forms

dxxfxgxgxf ))(')()(')((

dxexR xP )()(

( )P x dx

dxxxP ))sin(),(cos(

dxcbxaxxR ),( 2

dxexP x ),(

FORM EXAMPLE

dxx 2

dxex

x453

1

3(cos ( )sin( ) 4cos( ))x x x dx

dxex x 2

dxxxx 16522

dxexx x )25( 43

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.

Conclusions

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