MPSI - Dynnielux.dyndns.info/mpsi/Resumes/10-primitives.pdf · Lycée Robespierre Arras MPSI...
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Lycée Robespierre
Arras
MPSI
Primitives usuelles
Fonction Intervalles Primitives
(x 7→ . . .) (x 7→ . . .)
xα (α ∈ R, α ̸= −1) ]0,+∞[xα+1
α+ 1+ C
1
x]−∞, 0[ ou ]0,+∞[ ln |x|+ C
eax (a ∈ R∗) Reax
a+ C
ax (a ∈ ]0, 1[ ∪ ]1,+∞[) Rax
ln(a)+ C
sin(ax) (a ∈ R∗) R −cos(ax)
a+ C
cos(ax) (a ∈ R∗) Rsin(ax)
a+ C
sh(ax) (a ∈ R∗) Rch(ax)
a+ C
ch(ax) (a ∈ R∗) Rsh(ax)
a+ C
tan(x) =sin(x)
cos(x)
]−π
2+ kπ,
π
2+ kπ
[, (k ∈ Z) − ln |cos(x)|+ C
cotan(x) =cos(x)
sin(x)]kπ, (k + 1)π[, (k ∈ Z) ln |sin(x)|+ C
th(x) =sh(x)
ch(x)R ln (ch(x)) + C
coth(x) =ch(x)
sh(x)]−∞, 0[ ou ]0,+∞[ ln |sh(x)|+ C
1
sin(x)]kπ, (k + 1)π[, (k ∈ Z) ln
∣∣∣tan(x2
)∣∣∣+ C
1
cos(x)
]−π
2+ kπ,
π
2+ kπ
[, (k ∈ Z) ln
∣∣∣tan(x2+
π
4
)∣∣∣+ C
1
sh(x)]−∞, 0[ ou ]0,+∞[ ln
∣∣∣th(x2
)∣∣∣+ C
1
ch(x)R 2 arctan (ex) + C
1
a2 + x2(a > 0) R
1
aarctan
(xa
)+ C
1
a2 − x2(a > 0) ]−∞,−a[ ou ]−a, a[ ou ]a,+∞[
1
2aln
∣∣∣∣x+ a
x− a
∣∣∣∣+ C
1√a2 − x2
(a > 0) ]−a, a[ arcsin(xa
)+ C
1√a2 + x2
(a > 0) R ln∣∣∣x+
√a2 + x2
∣∣∣+ C
1√x2 − a2
(a > 0) ]a,+∞[ ln(x+
√x2 − a2
)+ C