Integrarea funcţiilor raţionale simple

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Definiţie: Funcţiile raţionale simple sunt de forma:
definite pe domeniile lor maxime


1
x  a
dx 
dx  ln x  a  C
I. 1)
xa
xa

1
1 ax  b 
1
dx 
dx  ln ax  b  C
2)
ax  b a ax  b
a
 n 1
1
1
1
n
 x  a 
dx  x  a   x  a  dx 
C  
C
II. 1)
n
n 1
 n 1
n  1 x  a 
x  a 






1
1
 n 1
 1 ax  b
dx  ax  b   ax  b  dx  
C 
2)
n
a  n 1
ax  b a
1
1


C
n1
an 1 ax  b


 n 1
III.
Caz
I

1)
1
dx , unde a  0 .
2
ax  bx  c
  0  ax 2  bx  c  ax  x1   I 
2
1
a

1
1 1
dx


C
2
a x  x1
x  x1 
(formula III. 1))
2

b 
 

2
Caz 2)   0  ax  bx  c  a  x    2  (forma canonică)
 2a  4a 

b


b

x 
x 
1
1
1
2a 

2a 2a  C 
I 
dx


ln
2
2
a 
a

b

b   
2
x 

 x    

2a
2a 2a
 2a   2a 

b

1
2a

ln

b
x 
2a
x

2a  C

2a
Caz 3)   0  din forma canonică

 b
b
x



x
1
1 1
1
2ax  b
2a 

2
a
I
dx  
arctg
C 
arctg
C
2
2
a  b    
a 




 x    
2a
2a
 2a   2a 

Observaţie: În cazul 2) avem:   0  ax2  bx  c  ax  x1 x  x2  
1 A
B 
 (se foloseşte formula I.1))

 

ax  x1 x  x2  a  x  x1 x  x2 
1

IV. Se ştie că ax 2  bx  c   2ax  b 
B
2aB
2aB
x
2ax 
2ax  b 
b
Ax  B
A
A
A dx 
A dx 
A
dx

A
dx 
2
2
2
2
ax  bx  c
ax  bx  c
2a ax  bx  c
2a
ax  bx  c
2aB

b
2


A
2ax  b
A
A
ax

bx

c
A  2aB 
A

dx 
dx 
dx   
 b 
2
2
2
2a ax  bx  c
2a ax  bx  c
2a ax  bx  c
2a  A

1
A
2aB  Ab
1
2

dx  ln ax  bx  c 

dx , unde
2
2
ax  bx  c
2a
2a
ax  bx  c
1
J
dx se calculează cu formula III.
2
ax  bx  c










V.
1 a2  x2  a2
1 a2  x2
dx  2
dx  2
dx 
2
2 n
2
2 n
2
2 n
a
a x  a 
x  a 
x  a 
1
x2
1
1
1
x 2  a 2 n  x 2  a 2   dx  1 I 
 2
dx

dx

x

n 1


a x 2  a 2 n
a 2 x 2  a 2 n1
2a 2
a2

 n 1
 n 1
2
2  n 1 




1
x a 
1
1  x 2  a 2 
x2  a2 
x
 x 
dx 
 dx  2 I n1  2  x 
2
2a
a
2a 
 n 1
 n 1
  n  1 

1
1
x
1
1
 I n  2 I n1  2 
 2 2 n1  2
I n1 
a
2a n  1 x  a 
2a n  1
1
In 
dx

x 2  a 2 n a 2


1

a2



2n  3
x
,
I

n 1
2
2
2
2 n 1
2a n  1
2a n  1x  a 
recurenţă din care se poate calcula I n .
 In 



ceea ce reprezintă relaţia de
VI. I n  
ax
Ax  B
2
,
cu


b
 4ac  0
dx
n
 bx  c 
Observaţie: Pentru   0  ax2  bx  c  ax  x1 x  x2  şi după descompunerea
2
în fracţii raţionale simple devine formula II..
B
A
A
In  A
dx

2a
ax 2  bx  c n
x


A
2a
 ax
2ax  b
2
 bx  c 
n

2aB
2aB
2ax  b 
b
A
A
A
dx 
dx 
n
n
2
2
2a
ax  bx  c 
ax  bx  c 
2ax 

2aB
b
A
A
A
dx 
dx

2a ax 2  bx  c n
2a


 ax
2
n

 bx  c   ax 2  bx  c  dx 

 n 1
2
A  2aB 
1
A
ax

bx

c
A  2aB 
 
 b
dx

n

I



  b  Jn
n
2a  A
 ax 2  bx  c 
2a
 n 1
2a  A 

Aplicaţii

1
1
1
1
2
4
1
dx;
dx;
dx;
dx;
dx;
dx; 2
dx;
2
5
5
x 1 2x  3
3 x
x  2 3  x 3x  1 x  2x  3

1
1
2x  1
1
x2
dx; 2
dx; 2
dx; I 2  2 2 dx; J 2  2
dx
2
2
x  4x  5
9x  6x  2
x  x 1
x  1
x  2x  5









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