Table des matières

10−2
I=Z1
0−x
2−xx
f:x7−→ f(x) =
−x
2−x[0 ; 1]
x[0 ; 1] 16f(x)61
2
J K J =Z1
0
(2 + x)−xx K =Z1
0
x2f(x)x
J= 3 −4
f(x)1
36K61 6
J+K= 4I
I
10−2I
(O, −→ ı , −→ )
fR
f(x) = x2−x.
f0f
f−∞ +∞
f0(x)f
fR
a I(a) = Za
0
f(x)x
a I(a)
a
I(a) = 2 −2−a1 + a+a2
2.
a
1
2
aI(a) = a−1 + a+a2
2.
g h Rg(x) = xh(x) = 1 + x+x2
2
CgPh
C P 0
C P

a b a < b f g [a;b]
•Zb
a
[f(t) + g(t)] t=Zb
a
f(t)t+Zb
a
g(t)t
•t∈[a;b], f(t)>0Zb
a
f(t)t>0
t∈[a;b], f(t)6g(t)Zb
a
f(t)t6Zb
a
g(t)t
n fn[0 ; +∞[
fn(x) = ln (1 + xn)
In=Z1
0
ln (1 + xn)x
Cnfn(O, −→ ı , −→ )
f1+∞
f1[0 ; +∞[
I1
I1
x∈[0 ; 1],x
x+ 1 = 1 −1 x+ 1
n06In6ln 2
(In)
(In)
g[0 ; +∞[
g(x) = ln(1 + x)−x.
g[0 ; +∞[
g[0 ; +∞[n
x
ln (1 + xn)6xn.
(In)
f[0 ; +∞[
f(x) = ln x2+ 4.
f[0 ; +∞[
g[0 ; +∞[g(x) = f(x)−x

g[0 ; +∞[
[2 ; 3] g(x) = 0
α
α−1
α f(x) = x
(un)u0= 1 n un+l=f(un)
Cf∆y=x
u0C∆u1
u2u3
−→
ı
−→
u0u1
ICα
n16un6α
(un)
f x
f(x) = 1 + xln x.
f0f
Cf(O, −→ ı , −→ )
Ty=x
C T

lim
x→0f(x)=1
xln x x ∈]0 ; 1]
f(x)61
f0(x)x∈]0 ; 1]
T C
g x ∈]0 ; 1]
g(x) = 1 + xln x−x.
g g
g0
C T
α0< α < 1
I(α) = Z1
α
[1 −f(x)] x.
I(α) = α2
2ln α+1 4−α2 4
lim
α→0I(α)
C T
fR
f(x) = 9
2
−2x−3−3x.
y0+ 2y= 3 −3x
0y0+ 2y= 0
hRh(x) = 9
2
−2x0
gRg(x) = −3−3x
f=g+h f
Cff(O, −→ ı , −→ )
xRf(x) = 3 −2x3
2−−x
f+∞f−∞
f f
Cf
f(1) Cf