TAVOLA DEGLI SVILUPPI DI TAYLOR DELLE FUNZIONI

TAVOLA DEGLI SVILUPPI DI TAYLOR DELLE FUNZIONI ELEMENTARI PER x → 0.
ex
=
sin x =
cos x =
tan x =
sinh x =
cosh x =
tanh x =
1
1−x
x2
x3
xn
+
+ ··· +
+ o(xn )
2
6
n!
x5
(−1)n 2n+1
+
+ ··· +
x
+ o(x2n+2 )
5!
(2n + 1)!
x4
(−1)n 2n
+
+ ··· +
x + o(x2n+1 )
4!
(2n)!
2
17 7
62 9
+ x5 +
x +
x + o(x10 )
15
315
2835
x5
x2n+1
+
+ ··· +
+ o(x2n+2 )
5!
(2n + 1)!
x4
x2n
+
+ ··· +
+ o(x2n+1 )
4!
(2n)!
2
17 7
62 9
+ x5 −
x +
x + o(x10 )
15
315
2835
1+x+
x3
6
x2
1−
2
x3
x+
3
x3
x+
6
x2
1+
2
x3
x−
3
x−
= 1 + x + x2 + x3 + · · · + xn + o(xn )
x2
x3
(−1)n+1 n
+
+ ··· +
x + o(xn )
2
3
n
x3
x5
(−1)n 2n+1
arctan x = x −
+
+ ··· +
x
+ o(x2n+2 )
3
5
2n + 1
x3
x5
x2n+1
arctanh x = x +
+
+ ··· +
+ o(x2n+2 )
3
5
2n + 1
µ ¶
α(α − 1) 2 α(α − 1)(α − 2) 3
α n
(1 + x)α = 1 + αx +
x +
x + ··· +
x + o(xn )
n
2
6
log(1 + x) =
con
x−
µ ¶
α(α − 1)(α − 2) · · · · · (α − n + 1)
α
=
n
n!
1
TAVOLA DI PRIMITIVE DI FUNZIONI ELEMENTARI
Z
xa dx
Z
Z
1
dx
x
ax dx
=
xa+1
+C
a+1
se a 6= −1
= log |x| + C
=
Z
ax
+C
log a
cos x dx
= sin x + C
sin x dx
= − cos x + C
Z
Z
1
dx
cos2 x
Z
1
dx
2
sin
x
Z
cosh x dx
Z
sinh x dx
= tan x + C
= − cotan x + C
= sinh x + C
= cosh x + C
Z
1
dx
2
Z cosh x
1
dx
2
sinh
x
Z
1
dx
2
a + x2
Z
1
√
dx
1 − x2
Z
1
√
dx
2
x +1
Z
1
√
dx
2
x −1
= tanh x + C
= − cotanh x + C
=
1
x
arctan + C
a
a
= arcsin x + C
³
´
p
= arcsinh x + C = log x + x2 + 1 + C
³
´
p
= arccosh x + C = log x + x2 − 1 + C
2
per x > 1