174
[A] Primitives usuelles
dt
sin
2 t
= − cot t + k
R \ {kπ} k ∈ Z
dt
sin t
= ln
tan
t
2
+ k
R \ {kπ} k ∈ Z
cot t dt = ln | sin t| + k
R \ {kπ} k ∈ Z
sh t dt = ch t + k
R
dt
sh
2 t
= − coth t + k
R ∗
dt
sh t
= ln
th
t
2
+ k
R ∗
coth t dt = ln | sh t| + k
R
a
t dt =
a t
ln a
+ k, (a ∈ R
∗
+ − {1})
R
Dans la suite on suppose : a ∈ R ∗
dt
t 2 + a 2 =
1
a
Arctan
t
a
+ k
R
dt
√
a 2 − t 2
=
Arcsin
t
|a|
+ k
− Arccos
t
|a|
+ k
] − a, a[
dt
√
t 2 + a 2
=
Argsh
t
|a|
+ k
ln
t +
√
t 2 + a 2
+ k
R
dt
√
t 2 − a 2
=

        
        

 
 
Argch
t
|a|
+ k
ln
t +
t 2 − a 2
+ k
]|a|, +∞[

 
 
− Argch
t
a
+ k
ln
t +
t 2 − a 2
+ k
] − ∞, |a|[
dt
√
t 2 + b
= ln
t +
t 2 + b
+ k, (b ∈
R
∗ )
R [−b, b]
dt
t 2 − a 2 =



Argth t + k
] − 1, 1[
1
2a
ln
t − a
t + a
+ k
R − {−a, a}