Réponses
algébriques aux sujets A,B,C,D de la seconde interrogation
avec Maxima
Maxima 5.12.0
http://maxima.sourceforge.net
...
Sujet
A, Sujet B, Sujet C,
Sujet D
Sujet
A - Ex.1
(%i1) integrate(1/(y-x)^2,y);
1
(%o1)
- -----
y - x
(%i2) subst(3,y,%o1)-subst(1,y,%o1);
1 1
(%o2)
----- - -----
1 - x 3 - x
(%i3) integrate(%o2,x);
(%o3)
log(3 - x) - log(1 - x)
(%i4) integrate(%o2,x,0,1/2);
5
(%o4)
- log(3) + log(-) + log(2)
2
(%i5) logexpand:super$expand(%o4);
(%o6)
log(5) - log(3)
Sujet A - Ex.2
(%i7) integrate(2/sqrt(x),x);
(%o7)
4 sqrt(x)
(%i8)
expand(integrate(2/sqrt(x),x,1,8));
(%o8)
8 sqrt(2) - 4
(%i9)
expand(integrate(x^(1/3)*log(x),x));
4/3
4/3
3 x log(x) 9 x
(%o9)
------------- - ------
4
16
(%i10) integrate(x^(1/3)*log(x),x,1,8);
135
(%o10)
12 log(8) - ---
16
(%i11)
%o8+%o10;
199
(%o11)
12 log(8) + 8 sqrt(2) - ---
16
(%i12) expand(integrate(x^(1/3)*log(x)+2/sqrt(x),x,a,8));
Is a - 8 positive, negative, or zero?
negative;
Is a positive, negative, or zero?
positive;
4/3
4/3
3 a log(a)
9 a
(%o12) - ------------- + ------ - 4 sqrt(a) + 12 log(8) + 8
sqrt(2) - 9
4
16
(%i13) subst(0,a,%);
log(0) has been generated.
-- an error. To debug this try debugmode(true);
(%i14)
limit(%o12,a,0,plus);
(%o14)
12 log(8) + 8 sqrt(2) - 9
Sujet A - Ex.3
(%i15) 'integrate((x+1)^2/sqrt(2-x),x,0,a);
a
/
2
[ (x + 1)
(%o15)
I ----------- dx
] sqrt(2 - x)
/
0
(%i16) changevar(%,t-(2-x),t,x);
2 - a
/ 2
[ t - 6 t +
9
(%o16)
- I ------------ dt
]
sqrt(t)
/
2
(%i17) expand(%);
2 - a
/
[
3/2
9
(%o17)
- I (t - 6 sqrt(t) +
-------) dt
]
sqrt(t)
/
2
(%i18) ev(%,nouns);
Is a positive, negative, or zero?
positive;
Is a - 2 positive, negative, or zero?
negative;
2
sqrt(2 - a) (2 a + 12 a + 58) - 58 sqrt(2)
(%o18) -
-------------------------------------------
5
(%i19) limit(%o17,a,2,minus);
58 sqrt(2)
(%o19)
----------
5
Sujet B - Ex.1
(%i1) integrate(1/(y-x-1)^2,y);
1
(%o1)
- ---------
y - x - 1
(%i2) subst(3,y,%o1)-subst(2,y,%o1);
1 1
(%o2)
----- - -----
1 - x 2 - x
(%i3) integrate(%o2,x);
(%o3)
log(2 - x) - log(1 - x)
(%i4) integrate(%o2,x,0,1/3);
5 2
(%o4)
- log(2) + log(-) - log(-)
3 3
(%i5) logexpand:super$expand(%o4);
(%o6)
log(5) - 2 log(2)
Sujet B - Ex.2
(%i7)
integrate(sqrt(x),x);
3/2
2 x
(%o7)
------
3
(%i8) integrate(sqrt(x),x,1,16);
(%o8)
42
(%i9) expand(integrate(x^(1/4)*log(x),x));
5/4
5/4
4 x log(x) 16 x
(%o9)
------------- - -------
5
25
(%i10) integrate(x^(1/4)*log(x),x,1,16);
128 log(16) 496
(%o10)
----------- - ---
5
25
(%i11) %o8+%o10;
128 log(16) 554
(%o11)
----------- + ---
5
25
(%i12) expand(integrate(x^(1/4)*log(x)+sqrt(x),x,a,16));
Is a - 16 positive, negative, or zero?
negative;
5/4
3/2 5/4
4 a log(a) 2
a 16
a 128
log(16) 1664
(%o12) -
------------- - ------ + ------- + ----------- + ----
5
3
25
5
75
(%i13) limit(%o12,a,0,plus);
1920 log(16) + 1664
(%o13)
-------------------
75
(%i14) expand(%);
128 log(16) 1664
(%o14)
----------- + ----
5
75
Sujet B - Ex.3
(%i15)
'integrate((x-1)^2/sqrt(3-x),x,0,a);
a
/
2
[ (x - 1)
(%o15)
I ----------- dx
] sqrt(3 - x)
/
0
(%i16) changevar(%,t-(3-x),t,x);
3 - a
/ 2
[ t - 4 t +
4
(%o16)
- I ------------ dt
]
sqrt(t)
/
3
(%i17) expand(%);
3 - a
/
[
3/2
4
(%o17)
- I (t - 4 sqrt(t) +
-------) dt
]
sqrt(t)
/
3
(%i18) ev(%,nouns);
Is a positive, negative, or zero?
positive;
Is a - 3 positive, negative, or zero?
negative;
2
sqrt(3 - a) (6 a + 4 a + 54) - 54 sqrt(3)
(%o18) -
------------------------------------------
15
(%i19) limit(%o17,a,3,minus);
18 sqrt(3)
(%o19)
----------
5
Sujet C - Ex. 1
(%i1) integrate(1/(y-x-1)^2,y);
1
(%o1)
- ---------
y - x - 1
(%i2) subst(4,y,%o1)-subst(2,y,%o1);
1 1
(%o2)
----- - -----
1 - x 3 - x
(%i3) integrate(%o2,x);
(%o3)
log(3 - x) - log(1 - x)
(%i4) integrate(%o2,x,0,1/2);
5
(%o4)
- log(3) + log(-) + log(2)
2
(%i5) logexpand:super$expand(%o4);
(%o6)
log(5) - log(3)
Sujet C - Ex.2
(%i7) integrate(3/sqrt(x),x);
(%o7)
6 sqrt(x)
(%i8) expand(integrate(3/sqrt(x),x,1,8));
(%o8)
12 sqrt(2) - 6
(%i9) expand(integrate(x^(1/3)*log(x),x));
4/3
4/3
3 x log(x) 9 x
(%o9)
------------- - ------
4 16
(%i10) integrate(x^(1/3)*log(x),x,1,8);
135
(%o10)
12 log(8) - ---
16
(%i11) %o10-%o8;
39
(%o11)
12 log(8) - 12 sqrt(2) - --
16
(%i12) expand(integrate(x^(1/3)*log(x)-3/sqrt(x),x,a,8));
Is a - 8 positive, negative, or zero?
negative;
Is a positive, negative, or zero?
positive;
4/3
4/3
3 a
log(a) 9 a
(%o12) - ------------- + ------ + 6 sqrt(a) + 12 log(8) - 12 sqrt(2) - 9
4 16
(%i13) limit(%o12,a,0,plus);
(%o13)
12 log(8) - 12 sqrt(2) - 9
Sujet C - Ex.3
(%i14)
'integrate((x-1)^2/sqrt(2-x),x,0,a);
a
/ 2
[ (x - 1)
(%o14)
I ----------- dx
] sqrt(2 - x)
/
0
(%i15) changevar(%,t-(2-x),t,x);
2 - a
/ 2
[ t - 2 t + 1
(%o15)
- I ------------ dt
] sqrt(t)
/
2
(%i16) expand(%);
2 - a
/
[
3/2
1
(%o16)
- I (t - 2 sqrt(t) +
-------) dt
]
sqrt(t)
/
2
(%i17) ev(%,nouns);
Is a positive, negative, or zero?
positive;
Is a - 2 positive, negative, or zero?
negative;
2
sqrt(2 - a) (6 a - 4 a + 14) - 14 sqrt(2)
(%o17) -
------------------------------------------
15
(%i18) limit(%,a,2,minus);
14 sqrt(2)
(%o18)
----------
15
Sujet D - Ex.1
(%i1) integrate(1/(y-x+1)^2,y);
1
(%o1)
- ---------
y - x + 1
(%i2) subst(2,y,%o1)-subst(0,y,%o1);
1 1
(%o2)
----- - -----
1 - x 3 - x
(%i3) integrate(%o2,x);
(%o3)
log(3 - x) - log(1 - x)
(%i4) integrate(%o2,x,0,1/2);
5
(%o4)
- log(3) + log(-) + log(2)
2
(%i5) logexpand:super$expand(%o4);
(%o6)
log(5) - log(3)
Sujet D - Ex.2
(%i7) integrate(1/sqrt(x),x);
(%o7)
2 sqrt(x)
(%i8) expand(integrate(1/sqrt(x),x,1,8));
(%o8)
4 sqrt(2) - 2
(%i9) expand(integrate(x^(2/3)*log(x),x));
5/3
5/3
3 x log(x) 9 x
(%o9)
------------- - ------
5 25
(%i10) integrate(x^(2/3)*log(x),x,1,8);
96 log(8) 279
(%o10)
--------- - ---
5 25
(%i11) %o10-%o8;
96
log(8)
229
(%o11)
--------- - 4 sqrt(2) - ---
5
25
(%i12) expand(integrate(x^(2/3)*log(x)-1/sqrt(x),x,a,8));
Is a - 8 positive, negative, or zero?
negative;
Is a positive, negative, or zero?
positive;
5/3
5/3
3 a
log(a) 9
a
96
log(8)
288
(%o12) - ------------- + ------ + 2 sqrt(a) + --------- - 4 sqrt(2) -
---
5
25
5
25
(%i13) limit(%o12,a,0,plus);
480 log(8) - 100 sqrt(2) - 288
(%o13)
------------------------------
25
(%i14) expand(%);
96
log(8)
288
(%o14)
--------- - 4 sqrt(2) - ---
5
25
Sujet D - Ex.3
(%i15)
'integrate((x+1)^2/sqrt(1-x),x,0,a);
a
/ 2
[ (x + 1)
(%o15)
I ----------- dx
] sqrt(1 - x)
/
0
(%i16) changevar(%,t-(1-x),t,x);
1 - a
/ 2
[ t - 4 t + 4
(%o16)
- I ------------ dt
] sqrt(t)
/
1
(%i17) expand(%);
1 - a
/
[
3/2
4
(%o17)
- I (t - 4 sqrt(t) +
-------) dt
]
sqrt(t)
/
1
(%i18) ev(%,nouns);
Is a positive, negative, or zero?
positive;
Is a - 1 positive, negative, or zero?
negative;
2
sqrt(1 - a) (6 a + 28 a + 86) - 86
(%o18)
- -----------------------------------
15
(%i19) limit(%,a,1,minus);
86
(%o19)
--
15