Calcul des Primitives et Intégrales - Cours

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a < b
f Ox x =a
x=b Oy
f≥0f[a, b]
f≤000−00
f
f[a, b]
b
Z
a
f(x)dx ,
b
Z
a
f(t)dt , ou
b
Z
a
f(u)du.
[a, b]n
b
Z
a
f(x)dx Rnh

b−a
n
Rn=hf(a+ 0 ·h) + hf(a+ 1 ·h) + ··· +hf(a+ (n−1) ·h)) =
h[f(a) + f(a+h) + ··· +f(a+ (n−1) ·h)] = h
n−1
X
k=0
f(a+kh)
f f0
M0
|I−Rn| ≤ n·hM0
2·h≤(b−a)2 2nM0
RnI n
RnI
lim
n→∞
b−a
n
n−1
X
k=0
fa+kb−a
n=
b
Z
a
f(x)dx

f(t)g(t)a<b<c
b
Z
af(t) + g(t)dt =
b
Z
a
f(t)dt +
b
Z
a
g(t)dt.
b
Z
a
k·f(t)dt =k·
b
Z
a
f(t)dt
k∈R
a < b
a
Z
b
f(t)dt =−
b
Z
a
f(t)dt
a b c
c
Z
a
f(t)dt =
b
Z
a
f(t)dt +
c
Z
b
f(t)dt
f≤g⇒
b
Z
a
f(t)dt ≤
b
Z
a
g(t)dt
b
Z
a
cl dt =l·(b−a)

m≤f(x)≤M⇒m≤1
b−a
b
Z
a
f(x)dx ≤M
1
b−a
b
Z
a
f(x)dx
f[a, b]
b
Z
a
f(x)dx
≤
b
Z
a|f(x)|dx
f g [a, b]a < b g
[a, b]ξ]a, b[
b
Z
a
f(x)g(x)dx =f(ξ)
b
Z
a
g(x)dx
F F 0=f
F f
f[a, x]
F
F(x) =
x
Z
a
f(t)dt
f
b
Z
a
F0(t)dt =F(b)−F(a)