Formulario completo Analisi 1

Formattazione del testo

R∈ −, = + = (x)), (x)g (x) =Derivate : arctan cos cos Per partiI n I x c g( f fα βN, 1n−10d 0(x)α α−1 f=x Rα−βα+β −(x)g(x) (x)g(x)dx−2 f fRαx ) )| sin(sin(= (x)| +logdx f cdx 2 2(x)fd x x Limiti notevoli Successioni= Formule parametricheln(a)aa sin(2x)Rdx = log(cos(x))dx  →+∞ > 1aπ1 cos(2x) =d Posto tan si ha :t(x) =log 2 adx ln(a)x sin(2x) R R 2t= = tan(x) log(cos(x))dx → =1 1a=sin x 1d cos(2x)= nln(x) 2 ∗ =1+t lim an→∞dx x McLaurin delle funzioni elementari → −12 < <0 1a1−t|x| =cos xd |x| = 2 3 nx x xx 2 ◦e · · · 1+t= + + + + + +1 xdx x 6 ∃ → ≤ −1a2! 3! n! 2td x x =tan x=e e n )o(x 21−tdx ( →+∞ > 0b→d Relazione Asintoto Tutti per 0a=sin(x) cos(x) α(α−1) 2 n b◦(1 α ++ = + +1 xx) ∗ =lim nαxdx

n→∞2 → +∞con n → <0 0bd −=cos(x) sin(x) α(α−1)·...·(α−n+1) n n· · · + + )x o(xdx ≈)sin(a an| ∗ ) =lim sin(x 0n n1d 2= = + (x)tan(x) 1 tan n→∞ n1 2 3 n 222 ◦ − · · ·dx = + + + + +1 x x x x(x)cos aa ∗ ) =lim cos(x 1≈ −− ≈ nn )) , cos(a 11 cos(a1+x n→∞ nnn1d 2 2n−=cot(x) )o(x x∗ =lim a n≈ ≈2 ) + )tan(a , log(1a a adx (x)sin n→∞n n n n2 3x x ◦ − · · ·d + = + + +log(1 x) x → ⇒≈ +∞ +∞)arctan(a xa=sinh(x) cosh x nn n2 3dx nxn+1 n ≈)sinh(a ad x+ )(−1) o(x → ⇒ ∈=cosh(x) sinh x x xa 0n n Rn 0ndx 2a3 51 2d x x − ≈ n)cosh(a 1 → ⇒ −∞= = + (x) ◦ − ···tanh(x) 1 tanh 0= + + + xarctan x x n n22dx 3 5(x)cosh a

₋ ≈1n ae²ⁿ⁺¹ ∗ (x ) =lim logx 2n+2n n1d (−1) + )o(x n→∞ n=arctan(x) aa2n+12dx ≈ +1e an1+x  → ⇒n +∞ +∞x³ 5x x1d n√ ◦ − · · ·= + + += sin x xarcsin(x) a − ≈1 log(x)anx 3! 5!dx √ n² → ⇒1−x (x ) >log 0x x2n+1 n0 0ne n ax 2n+2 ≈n )2πn(n!+ )(−1) o(x1d −=arccot(x) (2n+1)! + → ⇒0 0x2dx 1+x 1 nπ −≈ { ) =) arctan(arctan(a2 4x x n1d √ ◦ − · · ·= + + +cos 1 2x a− xn=arccos(x) −1n e∗ →=lim 1, 0x2 4!dx 1 n→∞ n2 π − →1−x = per a x2n ∞} nnx 2n+1n 2 a(−1) + )o(x nn 11d √ ∗ → →+1= e, nsettsinh(x) (2n)! ≈ ∞arcsin a a nn ndx 2 +1x 3 5 2n−1 a1x x x n◦ · ·+ → →∗− ≈ ·α= + +· + +(1 + )sinh 1x x+ e, a1a a_∞α_1d nn √ 3! 5!= a(2n-1)!settcosh(x) ndx x≈ ·α(1 + ) +1a a2 n-1 2n αx α∗ → α+1 seen n)o(x x1 nd =setttan(x) → ±∞, ∈2 4 2n xx x x α R2 ◦ ···dx n= + + + + +cosh x x1-x 2! 4! (2n)! Relazione asintoto1 1d −(arctan ) = 2n+12dx x 1)o(x1+x (x) =Se lim 0f → →∗(1 + ) 0e,x→x nε εε1 1d 0 nn−= Tavola Trigonometrica ≈(x)) (x)sin( f f2dx x x )log(1+xn →∗ = 1, 0lim x1 1+x 1d nn→∞(x)f=ln x− ≈ (x)1e f n22 1−xdx 1−x nε −1α∗ → → ∈ln(α), 0,x R1 2 α− ≈Integrali : ( (x))(x))1 fcos( f nεn2 α −1(1+x )α+1 ≈(x) (x)x arctan f f nR ∗α == + limx dx c αn→∞ xα+1 n≈+ (x)) (x)log(1 f f1 αR (a )|x|= +logdx c n∗ → 0x

1. (x)f− ≈ (x)1 log(a)a f ex x nR = +e dx e c α(a )n≈ →∗(x))
2. (x)tan( f f 0R −= + asin(x)dx cos(x) c a√ n≈(x))
3. (x)sinh( f f →n 1aR = +cos(x) sin(x) c ≈(x))
4. (x)tanh( f f1R = +tan(x)dx c2 c(x)cos − ≈ ·(1 + (x))
5. (x)1f c fR = +sinh coshxdx x c ≈(x))
6. (x)arcsin( f fR = +cosh sinhxdx x c 1 2− ≈(x))
7. (x)cosh( 1f fProprietà Potenze : 21R = +arctandx x c
8. Proprietà O-Piccoli :2 01+x α= =1, 1 1x1√R = +arcsin ±dx x c
9. (x) = (x))o( f o( f o( fa b a+b a a a· ·= = (xy),x x x x y21−x · ·(x)) = (x)) = (x))k o( f o(k f o( fa b a−b a a a/x = /y = (x/y),x x x1√R = + =settsinhdx x c √ · ·(x)) = (x))g(x) o( f o(g(x) f2 1+1x a b a·b√ n=(x ) = , xx x n√ · ·(x)) = (x))o(g(x)) o( f o(g(x) f2= + + +log(x 1)x c m n m=x xn1√ (x)) + (x)) = (x))o( f o( f o( fR = + =settcoshdx x c

Proprietà Modulo: |2 - 1x √(x))| = (x))o(o(f o(f|x| ≥ 0)2 - += + 1)log(x x c ≈ ⇔ (x) = (x) + g(x) f g(x) f|x| ≈ ⇔ | - |x| Limiti notevoli = = = 0 0, x x|R = + tan(x) log(1 - x) (x))o(fIntegrali notevoli: |x|x → Limiti notevoli per 0y|x · |x| · |y| = =, y| |y|yα+1(x)f0R α(x) (x)dx = + f f cα+1 Proprietà Logaritmi: 0 (x)fR |= (x)| + logdx f c (b)log = = log 1 0, a ba(x)f a0 · (x) (x)f f = + log(b log(b) log(c)c)R (x)dx = + e f e c bc 0 - = log log(b) log(c)R · - (x) (x)dx = (x)) + sin cos(f f f c0 c R · · = (x) (x)dx = (x)) + log log(b)cos sin(b cf f f c0R · (x) (x)dx = (x)) + Cambio base sinh cosh(f f f c0 ln(a)R · (x) (x)dx = (x)) + cosh sinh((a)f f f c logx→(b) = = log a b ba ln(b)0 (x)fR = (x) + arctandx f c → · = log log logx b x2 a a b(x))1+( f Trigonometria 0 (x)f√R → = (x) + arcsindx f c + = Per 0, lim log(1 - 0,x x) Formule di addizione e

sottrazione2(x))1−( f x =lim 1e0 ± ±(x)f ) =sin(α sin cos cos sin√ β α β α βR = (x) +settsinhdx f c Regole di derivazione2( (x)) +1f ± ∓) =cos(α cos cos sin sinβ α β α β 0 0(k (x)) = (x)Costante f k f0 (x)f√R tan= (x) +settcoshdx f c α±tan β± ) =tan(α 0 0 0β ± ±( (x) = (x) (x)Somma f g(x)) f g2 −1 1∓tan tan( (x))f α β 0 0Formule di duplicazione0p · ·( (x) = (x) +Prodotto2 f g(x)) f g(x)R − ·(x) (x)dx =* 1 f f = 0sin(2x) 2 sin(x) cos(x) ·(x) (x)f gp1 2−= [ (x) ( (x)) +1f f 222 −= (x) (x) =cos(2x) cos sin 0(x)f =Quoziente(x)] +arcsin f c 2 2 g(x)− −= (x) = (x)1 2 sin 2 cos 10p 0 02R ·(x) + (x)dx =* 1f f (x)·g(x)− (x)·g (x)f f=dalla precedente si ottiene 2 (x)gp1 2= [ (x) ( (x)) + +1f f 1−cos(2x)2 002 (x) =sin (x)f1 −