address: Faculty of Mathematics and Computer Science
"Babes-Bolyai" University
Kogalniceanu 1 Street
3400 Cluj-Napoca, Romania
The file INT_MISC.MTH contains three user functions which can be used to help or correct the integration process in DERIVE.
MAKECONT(y,x,v) := adds a locally constant function to y such that it becomes continuous at the points contained in the vector v.
CONT_INT(y, x, v) := INT(y,x), but it is "guaranteed" to be continuous at the points contained in the vector v. This function is useful in those (rare) occasions when DERIVE does not find a continuous antiderivative.
CONT_INT(y, x, v, a, b) := INT(y, x, a, b); the result is "guaranteed" to be correct even if the antiderivative found by DERIVE is not continuous at the points contained in the vector v.
For example, the antiderivative found by DERIVE for ASIN(2*x/(x^2+1)) is discontinuous at the points x=-1 and x=1.
To find a continuous antiderivative, use CONT_INT(ASIN(2*x/(x^2+1)), x, [-1,1]).
The correct integral for this function on the interval [0,2] is CONT_INT(ASIN(2*x/(x^2+1)), x, [-1,1], 0, 2) or just CONT_INT(ASIN(2*x/(x^2+1)), x, [1], 0, 2) (because -1 does not belong to the interval [0,2]).
INTSUBST(y, x, rx, t, rt) := the antiderivative of y(x) computed using the change of variable given by the relation rx.
rx is a relation (equality) between x and t which can be solved (by SOLVE) with respect to t.
If rx in not a relation but an expression depending on x, rx=t is considered.
t denotes the new (intermediate) variable, and rt is a relation equivalent to the relation rx (in a certain interval) which can be solved with respect to t. It was necesary to deal with two relations because the SOLVE function is not powerful enough.
If the relation rt is absent, rt:=rx.
As a bonus, the INTSUBST can apply on demand the well known Euler substitutions for y = R(x,SQRT(a*x^2 + b+x +c)), where R is a rational function: in this case rx must have one of the special forms
[As a recommandation, use "euler1" when a>0, "euler2" when c>0 and "euler3" when b^2-4ac>0].
Furthermore, if the second argument x is replaced with the vector [x,a,b], the definite integral is computed.
The user is responsible for the equivalence of the given relations in the interval where he/she wants to integrate.
As one can see, the INTSUBST is very versatile, and can be even extended for other changes of variable (in order to eliminate the computation by hand of the inverse substitution). Please inspect its definition in the file INT_MISC.MTH.
For example, if
f:= 1/(SQRT(x^2+x+4)+2*x+3)
then an antiderivative for f can be computed with:
