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// This file contains transformation rules suitable for finding
// integrals of expressions.
//
// At the moment, this is just a proof of concept. Further development
// will probably use the very nice set of rules derived by Albert D. Rich
// for computer-based evaluation of integrals:
//
// http://www.apmaths.uwo.ca/~arich/index.html
//
// It appears possible to use his machine-readable rules written for
// Mathematica and transform them automatically into Frink (some mathematical
// functions will of course need to be added.)
//
// Perhaps a better method will involve Risch integration which is very
// complex to implement but provides a decision procedure which can decide
// if an expression *can* be integrated.
//
transformations integrals
{
// Multiple integrals
// Bail-out condition for first integral
Integrate[_n, _x, 1] <-> Integrate[_n, _x]
// Otherwise take one integral and decrement.
Integrate[_n, _x, _times is isInteger] <-> Integrate[Integrate[_n,_x], _x, _times-1]
// Integrate both sides of an equation.
Integrate[_a === _b, _x] <-> Integrate[_a, _x] === Integrate[_b, _x]
// Expression entirely free of x
Integrate[_a, _x] :: freeOf[_a, _x] <-> _a _x
// Separate out multiplicative parts that are free of x
Integrate[_a _b, _x] :: freeOf[_a, _x] and expressionContains[_b, _x] <-> _a Integrate[_b, _x]
// Chain rules for sums
Integrate[_a + _b, _x] <-> Integrate[_a, _x] + Integrate[_b, _x]
// Powers of x
Integrate[_x^(_y:1), _x] :: _y != -1 <-> _x^(_y+1)/(_y+1)
// Special case of 1/x (also written as x^-1)
Integrate[_x^-1, _x] <-> ln[_x]
// b^(a x)
Integrate[_b^((_a:1) _x), _x] :: freeOf[_b,_x] and freeOf[_a, _x] <-> _b^(_a _x) / (_a ln[_b])
// ln[x]
Integrate[ln[_x], _x] <-> _x ln[_x] - _x
// Trigonometric functions
Integrate[sin[(_a:1) _x], _x] :: freeOf[_a,_x] <-> -1/_a cos[_a _x]
Integrate[cos[(_a:1) _x], _x] :: freeOf[_a,_x] <-> 1/_a sin[_a _x]
Integrate[tan[(_a:1) _x], _x] :: freeOf[_a,_x] <-> -1/_a ln[cos[_a _x]]
Integrate[sec[(_a:1) _x]^2, _x] :: freeOf[_a,_x] <-> 1/_a tan[_a _x]
Integrate[(1/cos[(_a:1) _x])^2, _x] :: freeOf[_a,_x] <-> 1/_a tan[_a _x]
Integrate[sin[_x] / (1-sin[_x]^2), _x] :: freeOf[_a,_x] <-> sec[_x]
Integrate[sin[_a _x] / (1-sin[_a _x]^2), _x] :: freeOf[_a,_x] <-> sec[_a _x]
// Definite integral.
// This is inefficient because it has to integrate the same expression twice.
// We need a way to define a variable and use that twice. And the
// substituteExpression is awkward, as it's not evaluated until an eval is
// called on the expression.. We need an operator form that is
// replaced at transformation time.
Integrate[_n, _x, _fromX, _toX] <-> substituteExpression[Integrate[_n, _x], _x, _toX] - substituteExpression[Integrate[_n, _x], _x, _fromX]
// Multiple definite integral
// Otherwise take one integral and decrement.
Integrate[_n, _x, _times is isInteger, _fromX, _toX] <-> substituteExpression[Integrate[_n, _x, _times], _x, _toX] - substituteExpression[Integrate[_n, _x, _times], _x, _fromX]
/* Integration by parts.
"When you apply integration by parts, there is
usually a choice of what to call u and what to call dv. The "LIATE"
heuristic provides a suggestion of how to do that. It doesn't always work,
but it works so often that it is worth remembering and using it as the
first attempt.
LIATE stands for:
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential
The heuristic says: when there are two factors in the integrand, choose u
as the one furthest to the left in LIATE and choose dv as the one furthest
to the right.
https://en.wikipedia.org/wiki/Integration_by_parts#LIATE_rule
TODO: Complete this. The problem is that if you choose poorly, this will
loop forever.
*/
// Note that this solves an integral twice.
// Integrate[_v _u, _x] <-> _u Integrate[_v, _x] - Integrate[D[_u, _x] * Integrate[_v, _x], _x]
}
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This is a program written in the programming language Frink.
For more information, view the Frink
Documentation or see More Sample Frink Programs.
Alan Eliasen was born 19967 days, 12 hours, 50 minutes ago.