integrales primitivesvid 2022-2023

Z
-500 0 500 1000 1500 2000
Eudoxe 400-355 Newton 1643-1729
Leibniz 1646-1716
Archimède 287-219 Cauchy 1789-1857
Riemann 1826-1866
Lebesgue 1875-1941
Galilée 1564-1642
Cavalieri 1598-1647
Pascal 1623-1662

f[a, b]
f[a, b]M x [a, b]f(x)6M
f[a, b]sup
x∈[a;b]
f(x)
f[a, b]M x [a, b]f(x)6M
x0∈[a, b]M=f(x0)max
x∈[a;b]f(x)
[a, b]
[a, b]
f[a;b]σ[a;b]
σ={x0=a < x1< ... < xn=b}
i∈ {1; 2; ...n}
mi=inf
x∈[xi−1;xi]
f(x)Mi=sup
x∈[xi−1;xi]
f(x)δ(σ) = max
i∈{1,2,...,n}xi−xi−1
f σ
s[a;b](f, σ) =
i=n
X
i=1
mi(xi−xi−1)
f σ
S[a;b](f, σ) =
i=n
X
i=1
Mi(xi−xi−1)

σ x0=a= 0 x1= 1,5x2= 2 x3= 4 x4= 5 = b
sup
x∈[x0;x1]
f(x)
f[a, b]
lim
δ(σ)→0S(f, σ)−s(f, σ)=0
Zb
a
f(x)dx
Zb
a
f(x)dx = lim
δ(σ)→0S(f, σ) = lim δ(σ)→0s(f, σ)
f[0,1] f(x)=1 x∈Q
ZΣ
f(x)dx f(x)Mimidx xi−xi−1δ(σ)
σ f [a, b]
Zb
a
f(x)dx = lim
n→+∞
n
X
k=1
f(a+kb−a
n)b−a n
f g [a;b]λ

Zb
a
(f+g) = Zb
a
f+Zb
a
g
Zb
a
λf =λZb
a
f
f>0 [a;b]Zb
a
f>0
g>f[a;b]Zb
a
g>Zb
a
f
Zb
a
f 6Zb
a|f|
f[a;b]M=1
b−aZb
a
f
∀c∈[a;b],Zb
a
f=Zc
a
f+Zb
c
f
Zb
a
fg2 6Zb
a
f2×Zb
a
g2

f:I→RIRF:I→Rf I
F I F 0=f
f, F, G :I→RF f I G −F=K K
f
u I R
f(x)F(x)u(x)∈...
u0uα, α 6=−1uα+1
α+ 1 Rα∈N,R∗
+α∈R−N
u0
uln |u|R∗
+R∗ −
u0cos usin uR
u0sin u−cos uR
u0tan u−ln |cos u|]−π
2;π 2[
u0eueuR
u0ch ush uR
u0sh uch uR
u0
1 + u2Arctan uR
u0
1−u2Argth u]−1; 1[
u0
√1−u2Arcsin u]−1; 1[
−u0
√1−u2Arccos u]−1; 1[
u0
√1 + u2Argsh uR
u0
√u2−1Argch u]1; +∞[
u0
cos2(u)tan u]−π 2;π 2[
u0
ch2uth uR
f1(x)=2xex2