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