cos(π/2 − t) = sin t, sin(π/2 − t) = cos t, tg (π/2 − t) = cotg t, cos( −t) = cos t, sin(−t) = − sin t, tg (−t) = −tg t.

1

Formule di trigonometria

Archi associati

cos(π + t) = − cos t, sin(π + t) = − sin t, tg (π + t) = tg t, cos(π − t) = − cos t, sin(π − t) = sin t, tg (π − t) = −tg t, cos(π/2 + t) = − sin t, sin(π/2 + t) = cos t, tg (π/2 + t) = −cotg t,

cos(π/2 − t) = sin t, sin(π/2 − t) = cos t, tg (π/2 − t) = cotg t, cos( −t) = cos t, sin(−t) = − sin t, tg (−t) = −tg t.

Formule di addizione, sottrazione, duplicazione

cos(t 1 − t 2 ) = cos t 1 cos t 2 + sin t 1 sin t 2 , cos(t ₁ + t ₂ ) = cos t ₁ cos t ₂ − sin t 1 sin t ₂ , sin(t 1 − t 2 ) = sin t 1 cos t 2 − sin t 2 cos t 1 , sin(t ₁ + t ₂ ) = sin t ₁ cos t ₂ + sin t ₂ cos t ₁ , tg (t ₁ + t ₂ ) = tg t 1 + tg t 2

1 − tg t 1 tg t ₂ , tg (t 1 − t 2 ) = tg t 1 − tg t 2

1 + tg t 1 tg t 2

, sin 2t = 2 sin t cos t , cos 2t = cos ² t − sin ² t ,

tg 2t = 2tg t 1 − tg ² t .

Formule di Werner e di prostaferesi

sin t 1 cos t 2 = 1

2 [sin(t 1 + t 2 ) + sin(t 1 − t 2 )] , sin t 1 sin t 2 = 1

2 [cos(t 1 − t 2 ) − cos(t 1 + t 2 )] , cos t 1 cos t 2 = 1

2 [cos(t 1 − t 2 ) + cos(t 1 + t 2 )] , sin p + sin q = 2 sin p + q

2 cos p − q 2 , sin p − sin q = 2 cos p + q

2 sin p − q 2 , cos p + cos q = 2 cos p + q

2 cos p − q 2 , cos p − cos q = −2 sin p + q

2 sin p − q 2 . Formule di bisezione 

sin t 2    

=

r 1 − cos t

2 ,

cos t 2    

=

r 1 + cos t

2 .

Formule parametriche

sin x = 2t

1 + t ² , cos x = 1 − t ²

1 + t ² dove t = tg x 2 Funzioni iperboliche: relazioni fondamentali

cosh ² x − sinh ² x = 1, cosh 2x = cosh ² x + sinh ² x sinh 2x = 2 sinh x cosh x

2 Tabella delle derivate fondamentali

f (x) f ⁰ (x)

x ⁿ nx ^{n −1} n ∈ N

x ^α αx ^{α −1} x ∈ (0, +∞)

a ^x a ^x log a a > 0

log _a x 1

x log a = 1 x log ^a e a > 0, a 6= 1

sin x cos x

cos x − sin x

tg x 1

cos ² x = 1 + tg ² x cotg x − 1

sin ² x = −1 − cotg ² x

arcsin x 1

p 1 − x ²

arccos x −1

p 1 − x ²

arctan x 1

1 + x ²

arcotg x - 1

1 + x ²

sinh x cosh x

cosh x sinh x

tanh x 1

cosh ² x = 1 − tanh ² x coth x − 1

sinh ² x = 1 − coth ² x

settsinh x 1

p 1 + x ²

settcosh x 1

p x ² − 1

setttanh x 1

1 − x ²

settcotg x 1

1 − x ²

Serie di Fourier

a ₀ +

∞

X

k=1

a _k cos k 2π

T x
+ b k sin k 2π T x
, dove a 0 = _T ¹ R T +c

c f (x)dx, a k = _T ² R T +c

c f (x) cos k ^2π _T x
dx, b k = _T ² R T +c

c f (x) sin k ^2π _T x
dx. Inoltre, l’identit`a di Parseval ` e

Z T +c c

f (x) ² dx = T a ² ₀ + T 2

X ∞ k=1

(a ² _k + b ² _k ).

Operazioni su campi vettoriali Se F = (f ₁ , f ₂ , f ₃ ) ` e un campo vettoriale di classe C ¹ ,

div F = ∂f 1

∂x + ∂f 2

∂y + ∂f 3

∂z ,

rot F = ∇ ∧ F =      

i j k

∂

∂x

∂

∂y

∂

∂z

f 1 f 2 f 3

=  ∂f ₃

∂y − ∂f 2

∂z



i +  ∂f ₁

∂z − ∂f 3

∂x



j +  ∂f ₂

∂x − ∂f 1

∂y



k.

3 Sviluppi di McLaurin

e ^x =

∞

X

k=0

x ^k

k! ( −∞, +∞)

sin x =

∞

X

k=0

( −1) ^k x ^2k+1

(2k + 1)! ( −∞, +∞) cos x =

∞

X

k=0

( −1) ^k x ^2k

(2k)! ( −∞, +∞)

sinh x = X ∞ k=0

x ^2k+1

(2k + 1)! ( −∞, +∞)

cosh x = X ∞ k=0

x ^2k

(2k)! ( −∞, +∞)

arctan x =

∞

X

k=0

( −1) ^k

2k + 1 x ^2k+1 [ −1, 1]

(1 + x) ^α =

∞

X

k=0

α k



x ^k ( −1, 1)

log(1 + x) =

∞

X

k=1

( −1) ^{k −1}

k x ^k ( −1, 1]

1 1 − x =

∞

X

k=0

x ^k ( −1, 1)

Equazioni differenziali 1.) y ⁰ = a(x)y + f (x)

Soluzione: y(x) = ke ^A(x) + e ^A(x) R x

x

e ^−A(s) f (s)ds, (A(x) ` e una primitiva di a(x)).

2.) y ⁰⁰ + by ⁰ + cy = 0 Soluzione:

a) y(x) = c 1 e ^m

^x + c 2 e ^m

^x , (se m 1 e m 2 sono le soluzioni reali e distinte di m ² + bm + c = 0);

b) y(x) = (c ₁ x + c ₂ )e ^mx , (se m ` e la soluzione doppia di m ² + bm + c = 0);

c) y(x) = c 1 e ^−bx/2 cos βx + c 2 e ^−bx/2 sin βx, (dove β = pc − b ² /4, se l’eq. m ² + bm + c = 0 non ha sol. reali).

Integrali

Z

x ^α dx = 1

α + 1 x ^α+1 + c se α 6= −1 Z dx

x = log |x| + c Z

e ^x dx = e ^x + c Z

a ^x dx = a ^x log a + c Z

sin x dx = − cos x + c Z

cos x dx = sin x + c

4 Z

tan x dx = − log |cos x| + c Z

cot x dx = log |sin x| + c

Z 1

cos x dx = log | 1

cos x + tan x | + c

Z 1

sin x dx = log | 1

sin x − cot x| + c

Z 1

a ² + x ² dx = 1

a arctan x a + c

Z 1

a ² − x ² dx = 1

2a log | x + a x − a | + c

Z 1

√ a ² − x ² dx = arcsin x a + c Z p

x ² ± a ² dx = x 2

p x ² ± a ² ± a ²

2 log |x + p

x ² ± a ² | + c

Z dx

√ x ² ± a ² = log |x + p

x ² ± a ² | + c Z p

a ² − x ² dx = x 2

p a ² − x ² + a ²

2 arcsin x a + c Z

sin(ax) sin(bx) dx = sin((a − b)x)

2(a − b) − sin((a + b)x)

2(a + b) + c, se a ² 6= b ² Z

cos(ax) cos(bx) dx = sin((a − b)x)

2(a − b) + sin((a + b)x)

2(a + b) + c, se a ² 6= b ² Z

sin(ax) cos(bx) dx = − cos((a − b)x)

2(a − b) − cos((a + b)x)

2(a + b) + c, se a ² 6= b ² Z

sin ⁿ x dx = − 1

n sin ^{n −1} x cos x + n − 1 n

Z

sin ^{n −2} x dx, se n ≥ 2 Z

cos ⁿ x dx = 1

n cos ^{n −1} x sin x + n − 1 n

Z

cos ^{n −2} x dx, se n ≥ 2 Z

tan ⁿ x dx = 1

n − 1 tan ^{n −1} x − Z

tan ^{n −2} x dx, se n ≥ 2 Z

cot ⁿ x dx = − 1

n − 1 cot ^{n −1} x − Z

cot ^{n −2} x dx, se n ≥ 2

Z 1

cos ⁿ x dx = 1 n − 1

1

cos ^{n −2} x tan x + n − 2 n − 1

Z dx

cos ^{n −2} se n ≥ 2

Z 1

sin ⁿ x dx = − 1 n − 1

1

sin ^{n −2} x cot x + n − 2 n − 1

Z dx

sin ^{n −2} se n ≥ 2 Z

sin ⁿ x cos ^m x dx = − sin ^{n −1} x cos ^m+1 x

n + m + n − 1 n + m

Z

sin ^{n −2} cos ^m x dx se n 6= −m Z

sin ⁿ x cos ^m x dx = sin ⁿ⁺¹ x cos ^{m −1} x

n + m + m − 1 n + m

Z

sin ⁿ x cos ^{m −2} x dx se n 6= −m Z

x ⁿ sin x dx = −x ⁿ cos x + n Z

x ^{n −1} cos x dx Z

x ⁿ cos x dx = x ⁿ sin x − n Z

x ^{n −1} sin x dx Z

arcsin x dx = x arcsin x + p

1 − x ² + c Z

arctan x dx = x arctan x − 1

2 log(1 + x ² ) + c.