poly de cours

C1
Γ

f[a, b]
Zb
a
fZ[a,b] fZb
a
f(t)dt t
a > b Zb
a
f(t)dt =−Za
b
f(t)dt Z[a,b] fZ[b,a] f
f
a<b Zb
a
f>0
f6g a 6bZb
a
f6Zb
a
g
g−f
f S = [a, b]
a=x0< x1··· < xn=b k ∈[[0, n −1]]
f]xk, xk+1[x+
kx− k+1
f]xk, xk+1[ [xk, xk+1]
f xkf
I[0,+∞[ ]0,1] Rf I
I

x7→ E(x) = bxc[0,3] R
x7→ (1
xsi x∈]0,2]
1 si x= 0 [0,1] x7→ (sin 1
xsi x∈]0,1]
0 si x= 0
]0,1]
f[a, b]
b−a
n
n−1
X
k=0
fa+kb−a
n−→
n→+∞Zb
a
f
n
X
k=1
k+n
k2+n2
Z1
0
x+ 1
x2+ 1dx =π 4+1 2ln 2

f I a ∈I x 7→ Zx
a
f(t)dt
C1f
Z1
0
f(t)dt Z√2
0
f(t)dt Zx
0
f(t)dt
x7→
Zx
a
f(x)dx
f f
x7→ Zx
a
f(t)dt f x 7→ Zx
a
f(t)dt
f x 7→ Zb
a
f(t)dt
f(x0)x0f x0
Ψ : x7→ Zβ(x)
α(x)
fΦ : x7→ Zx
a
f(t)dt) Ψ(x) = Φ (β(x)) −Φ (α(x))
f F f I a, b ∈IZb
a
f=F(b)−F(a)
Zb
a
t dt =
t2
2b
a
gC1I
∀a, b ∈I, g(b) = g(a) + Zb
a
g0(t)dt.
u v C1I a, b ∈I
Zb
a
u0(t)v(t)dt = [uv]b
a−Zb
a
u(t)v0(t)dt.
f∈ CM(J)ϕ∈ C1(I, J)a, b ∈J
Zϕ(b)
ϕ(a)
f(x)dx =Zb
a
f(ϕ(t)) ϕ0(t)dt.

x=ϕ(t)
dx =ϕ0(t)dt x =ϕ(t)ϕ(a)ϕ(b)t a b
ZB
A
f(x)dx t =ψ(x)
a b ψ(a) = A
ψ(b) = B ψ
dt =ψ0(x)dx dx =dt
ψ0(x)=··· ψ0
[0,1]
[0,1]
f>0 [a, b]a < b
fZb
a
f > 0
x0f(x0)>0α|f(x)−f(x0)|6f(x0)/2|x−x0|6α
f g f(x0)α > 0
f>0