Analisi I - Appunti/Riassunti

R

• → ∈ ̸ ∼

Se a ℓ con ℓ = 0 allora a ℓ

n n

• Utilizzo nel calcolo dei limiti: prodotti, quozienti, potenze, ma non nelle somme

• ∼ → ∈ ̸ ∼

Se a b e a , b ℓ [0, +∞] con ℓ = 1 allora log a log b

n n n n n n

α α α α

• ∼

Se p = c n + c n + . . . c n allora p c n

1 2 1

k

n 1 2 k n 1

R

∈ ̸

- qui c , c , . . . , c con c = 0 e α > . . . > α

1 2 k 1 1 k

- i coefficienti reali c , . . . c possono essere rimpiazzati da successioni limitate

2 k √

2 2

− ∼

- ad esempio 2n + (1 + cos n)n n 6 2n

→

Tabella (ε 0)

n ∼

1) sin ε ε

n n

∼

2) arcsin ε ε

n n

∼

3) tan ε ε

n n

∼

4) arctan ε ε

n n 2

ε

− ∼

5) 1 cos ε n

n 2

∼

6) sinh ε ε

n n

∼

7) tanh ε ε

n n 2

ε

− ∼

8) cosh ε 1 n

n 2

1/ε ∼

9) (1 + ε ) e

n

n

ε − ∼

10) e 1 ε

n n

∼

11.a) log(1 + ε ) ε

n n

∼ − →

11.b) log a a 1 [se a 1]

n n n R]

α − ∼ ∈

12.a) (1 + ε ) 1 αε [con α

n n

a − ∼ →

12.b) (1 + ε ) 1 a ε [se a ε 0]

n

n n n n n

√

−n n

∼

13) n! e n 2πn (de Moivre-Sterling)

∼

14) log n! n log n

n

! 1 ∼

15) log n (Eulero-Mascheroni)

k

k=1 1

SVILUPPI DI TAYLOR NOTEVOLI

2 3 n

x x x

= 1 + + + + + + )

x n

e x ... o(x

2! 3! n!

2 3 n

x x x

log(1 + = + + + (°1) + )

n+1 n

°

x) x ... o(x

2 3 n

3 5 2n+1

x x x

sin = + + + (°1) + )

n 2n+2

°

x x ... o(x

3! 5! 2n + 1!

2 4 2n

x x x

cos = 1 + + + (°1) + )

n 2n+1

°

x ... o(x

2! 4! 2n!

3 5 2n+1

x x x

arc tan = + + + (°1) + )

n 2n+2

°

x x ... o(x

3 5 2n + 1

1 1 3 (2n 1)!!

3 5 2n+1

· °

x x x

arc sin = + + + + + )

2n+2

x x ... o(x

2 3 2 4 5 (2n)!! 2n + 1

·

3 5 2n+1

x x x

sinh = + + + + + )

2n+2

x x ... o(x

3! 5! 2n + 1!

2 4 2n

x x x

cosh = 1 + + + + + )

2n+1

x ... o(x

2! 4! 2n!

2

3

x

tan = + + + )

5 6

x x x o(x

3 15 1) 1)(Æ 2)

° ° °

Æ(Æ Æ(Æ

(1 + = 1 + + + + )

Æ 2 3 3

x) Æx x x o(x

2! 3!

TABELLA INTEGRALI

0 dx c

∫ ⋅ =

dx x c k f ( x ) k f ( x )

dx

∫ ∫ ∫

= + ⋅ = ⋅ 1

n 1

x + n n 1

[ ] [ ] +

'

f ( x ) f ( x ) dx f ( x ) c

n

x dx c

, ( n 1) ∫ ⋅ = +

= + ≠ −

∫ n 1

+

n 1

+

dx f ' ( x )

dx x c dx f ( x

) c

= + = +

∫ ∫

2 x 2 f ( x

)

sinx dx cos x c sen f ( x ) f ' ( x

) dx cos f ( x

) c

∫ ∫

= − + ⋅ = − +

cos x dx sen x c cos f ( x ) f '( x )

dx sen f ( x ) c

∫ ∫

= + ⋅ = +

1 f ' ( x )

dx tg x c dx tg f ( x ) c

= +

∫ ∫ = +

2 2

cos x cos f ( x

)

1 f ' ( x )

dx ctg x c dx ctg f ( x ) c

= − +

∫ ∫ = − +

2 2

sen x sen f ( x )

dx f '( x )

arc sin x c dx arcsin f ( x ) c

∫ ∫

= + = +

2

2

1 x [ ]

1 f ( x )

− −

dx f '( x )

arctg x c dx arctg f ( x ) c

∫ ∫

= + = +

2

2

1 x [ ]

1 f ( x )

+ +

dx f '( x )

ln x c dx ln f ( x ) c

∫ ∫

= + = +

x f ( x )

x x f ( x ) f ( x )

e dx e c e f '( x ) dx e c

∫ ∫

= + = +

x f ( x )

a a

x f ( x )

a dx c a f ' ( x ) dx c

∫ ∫

= + = +

ln a ln a

m 1 n 1

( x a ) ( a bx )

+ +

+ +

m n

( x a ) dx c ( a bx

) dx c

∫ ∫

+ = + + = +

m 1 b ( n 1

)

+ +

dx 1 x dx 1

arctg c c

∫ ∫

= + = − +

a a

2 2 2

a x ( a bx

) b (

a bx )

+ + +

dx 1

n 1

( a bx ) +

+ c

n ∫

( a bx

) dx c = − +

∫ + = + 2

( a bx

) b (

a bx )

b ( n 1

) + +

+ 1 x

1 1 1 x

+ tg c

dx ln c ∫ = +

∫ = + 1 cos x 2

2

1 x 2 1 x +

− −

1 x tg x dx ln cos x c

∫ = − +

dx ctg c

∫ = − +

1 cos x 2

− dx x

ctg x dx ln sin x c

∫ = + ln tg c

∫ = +

sin x 2

dx 1 1 sinx

+ 2

arcsin x dx x arcsin x 1 x c

∫ = + − +

ln c

∫ = +

cos x 2 1 sinx

− 1

2

arccos x dx x arccos x 1 x c

∫ = − − + 2

arctg xdx x arctg x ln 1 x c

∫ = − + +

2

1 dx 1

2

arcctgxdx xarcctgx ln 1 x c ln a bx c

∫ ∫

= + + + = + +

2 a bx b

+

dx 1 b dx 1 ab bx

  +

arctg x c dx ln c

∫ ∫

 

= ⋅ + = +

2 2

a bx a a bx

ab 2 ab ab bx

+ − −

  dx x

2

x a x arcsin c

2 2 2 2

a x dx a x arcsin c ∫

∫ − = − + + = +

2 2 a a

2 2

a x

− 2

2

x a

2 2 2 2 2 2 3

a x dx a x ln x a x c a bx dx (

a bx ) c

∫ + = + + + + + ∫ + = + +

2 2 3

b

dx dx 2

2 2

ln x a x c a bx c

∫ ∫

= + ± + = + +

b

a bx

2 2

a x +

±

dx 1 x 1

− ln xdx x ln x x c

∫ = − +

ln c

∫ = +

2

x 1 2 x 1

− +

ln x ln x 1 1

2

dx c cos xdx ( x sinx cos x

) c

∫ ∫

= − − + = + +

x x 2

2

x 1

1 2

2 cos ( x a ) dx ( x sin ( x a ) cos( x a ) c

sin xdx ( x sinx cos x ) c ∫

∫ − = + − − +

= − + 2

2

dx x dx x

π

 

ln tg c ln tg c

∫ = + ∫ = −  −  +

sinx 2 cos x 4 2

 