Analiza

Derivate f ( x) − f ( x 0 ) ∃ lim = 0 şi este finită. x → x0 x − x0

f(x) are derivată în x0 dacă

1.

xn

2.

f ' ( x) 0 nx n −1 rx r −1 ex a x ln(a ) 1 1 ⋅ ⋅u' 2 u 1 x cos( x) − sin( x)

f (x) ctn (x ≥ 1∈ I )

No.

5.

xr ex ax

6.

u

3. 4.

7.

ln( x )

8. 9.

sin( x) cos( x )

10.

arcsin( x)

11.

arccos( x)

12.

arctg ( x )

13.

arcctg (x )

No.

Operands

Result

1.

( f + g ) ( x)

f ( x ) + g ' ( x)

2.

( fg ) ' ( x)

f ' ( x ) ⋅ g ( x ) + f ( x) ⋅ g ' ( x )

3.

(λ ⋅ f ( x)) '

4.

f ( ) ' ( x) g

λ ⋅ f ' ( x) f ' ( x) ⋅ g ( x ) − f ( x ) ⋅ g ' ( x ) g 2 ( x)

5.

(

f ( x)

f ' ( x)

'

λ ( g
f ) ' ( x)

6. Trick:

)'

a = e ln( a ) .

1 1− x2 −1

1− x2 1 1+ x2 −1 1+ x2

∀x ∈ (−1,1)

∀x ∈ (−1,1) ∀x ∈ R ∀x ∈ R

'

λ g ( f ( x)) ⋅ f ' ( x) '

Integrale Nedefinite No.

SpecificaŃii

Integrala nedefinită

f : R → R; 1.

2.

3.

f ( x) = x n ; n ∈ N f : J → R; J ⊂ (0, ∞) f ( x) = x a ; a ∈ R − {1} f :R→ R

n ∫ x dx =

x n +1 +c n +1

a ∫ x dx =

x a +1 +c a +1

ax ∫ a dx = ln(a) + c x

f ( x) = a x ; a ∈ R+* − {1} f : J → R; J ⊂ R *

4.

1 f ( x) = x f : J → R; J ⊂ R − {− a, a}

5.

1 f ( x) = 2 , (a ≠ 0) x − a2 f :R→R 1 f ( x) = 2 ;a ≠ 0 x + a2 f :R→ R f ( x ) = sin( x ) f :R→R f ( x) = cos( x)

6.

7.

8.

9.

1

∫ x dx = ln( x ) + c ∫x

2

∫x

2

1 1 x−a dx = ln +c 2 2a x + a −a

x 1 1 dx = arctg ( ) + c 2 a a +a

∫ sin( x)dx = − cos( x) + c ∫ cos( x)dx = sin( x) + c

π   f : J → R; J ⊂ R − (2k + 1) k ∈ Z  2   1 f ( x) = cos 2 ( x )

1

∫ cos

f : J → R; J ⊂ R − {kπ k ∈ Z } 10.

11.

12.

13.

f ( x) =

∫ sin

1 sin 2 ( x)

π   f : J → R; J ⊂ R − (2k + 1) k ∈ Z  2   f ( x) = tg ( x )

( x)

1 2

( x)

dx = tg ( x ) + c

dx = −ctg ( x) + c

∫ tg ( x)dx = − ln( cos( x) ) + c

f : J → R; J ⊂ R − {kπ k ∈ Z } f ( x) = ctg ( x) f : J → R; J ⊂ (−a, a), a > 0, 1 f ( x) = a2 − x2

2

∫ ctg ( x)dx = ln( sin( x) ) + c ∫

1

x dx = arcsin( ) + c a a2 − x2

f : R → R; a ≠ 0 1 f ( x) = x2 + a2 f : J → R, J ⊂ (−∞,−a ) sau J ⊂ (a, ∞ ), a > 0, 1 f ( x) = 2 x − a2

14.

15.

∫

∫

1 x +a 2

2

1 x −a 2

2

dx = ln( x + x 2 + a 2 ) + c

dx = ln x + x 2 − a 2 + c

Alte formule:

E x (x + 1) 2

2

2

=

A B Cx + D Fx + G + 2 + 2 + 2 x x x + 1 (x + 1) 2

Se notează:

ax 2 + bx + c = x a + t = xt + c

daca a > 0 daca c > 0

= (x - x 1 )t daca a < 0, c < 0, ∆ = b 2 − 4ac > 0 I.

x tg ( ) = t ⇔ x = 2arctg (t ) = ρ 2 2 ρ'= 1+ t 2 2t 1− t2 2t sin( x ) = , cos( x) = , tg ( x ) = 2 2 1+ t 1+ t 1− t2 II. R(-sin x,

cos x) = R(sin x, cos x) tg ( x) = t ⇒ x = arctg (t ) cos 2 ( x) =

1 t2 t 2 , sin ( ) = , sin( x ) cos( x ) = x 2 2 1+ t 1+ t 1+ t2

R(-sin x, cos x) = R(sin x, cos x) cos( x ) = t ⇒ x = arccos(t )

III.

∫ f ( x) g ' ( x)dx =

fg − ∫ f ' ( x) g ( x )dx