integrus

	Methods / ti-cas.org
	Integral calculus (C.A.S. v1.00)

Objectives	Resolution of 6 exercises posed at the time of the oral examinations of various contests of large scientific or commercial schools.
Public	Final Classes, Bts, Sup/Spe
Prérequis	Knowledge of the various methods of resolution
Idea	H.Lemberg

A LITTLE MATH
The primitive of a function f(x) is a F(x) function such as its derivative F' (x)=f(x). This operation is thus the opposite operation of that of derivation.
But if the calculation of a derivative is a purely technical operation (it is indeed enough to observe the rules of derivation of a sum, a product, a quotient...), the calculation of a primitive is more complex. Thus, the simple function f(x)=exp(x) admits primitives, but which can be expressed only using function elementary (radicals, exponential, functions trigo...).
In class, one use often some heuristic (some easy way...) like "change of variable" or the "integration by part" and one include easily that, if these technique be easy to put in application on a computer, the moment relevant of their use by the machine be much more delicate.
A LITTLE HISTORY
The first work on the calculation of primitives of algebraic functions goes back to 1830 with the mathematician Joseph LIOUVILLE. But it is only in the years 1970, with Ritt Rich, then 1980 with J Davenport which one developed calculation algorithms effective.
TI 92 integrates these algorithms.

Calculate (Central P')

Not less than 172 characters... for this primitive! (which does not hold on only one screen !)
TI 92 spent 3 small seconds to carry out this calculation...
Spectacular !

The integral is also obtained quickly:

Calculate (Ensi)

No problem for the TI-92.

Time to say "ouf", and the answer appears !

Let us pass to a more difficult exercise...

Calculate f(x) = (Po X Me)

Determine if F is a rational fraction and determine F then.

The TI-92 meets only few problems to solve this exercise.

The solution is:

Thus F is a rational fraction if and only if a=-10/3.

In this case,

Calculate f(x) = (Central P')

The TI-92 returns an approximate value, moreover rather quickly, which is the sign that it " refuses " to make a symbolic calculation.

It remains us to help it.

Like 0<=x<=1, let us pose x=sin(t).

Thus, =

This change of variable can be assisted by the program present on this site.

After a very long time very, one obtains this result .

One would like right knowledge how y to arrive.

One will be able obviously to write a programme of integration by parts.

Calculate f(x) = (X Me)

This is already a more delicate exercise. One should not hope to see TI-89.92 Plus (v1.00) calculating cete integral generalized.

Here, our computer will help us to ensure calculations, but it will be necessary to guide it and especially with the oral examination as with the writing, to explain its step, the theorems used and their validity within the framework of this exercise.

After have show that f(x) exist for any reality, that F be a function even (the field of study be thus [ 0, inf [), one show that F be derivable on ]0, inf ] and that:

Without " putting questions ", the TI-92 reasons like us.

The calculation of f' (x), for x>0, is then easy:

One knows whereas for x>0, f(x)=pi*ln(x-1)+cste
to give the value of this constant, let us calculate f(1).
The TI-92 gives only one approximate value of it.
Let us carry out then a change of variable t=tan(v):

The TI-92 finds only one value approximate.

It only remains to roll up the handles and to calculate:

this integral is one of most traditional among those studied in prépa and we advise you to retain this pretty sequence of change of variables which lead to the solution.

From where the value of I and f(1)=-Ì=pi*ln(2).

Therefore f(x)=pi*ln(1+x) for x>0.

The TI-92 lets to us suppose that f(0)=0 (see preceding screen).

Finally, f(x)=pi*ln(1+| x|)