Integrale Tabelare

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S.3.2-3. Integrals containing (

2 +

2)1 2.

18.

(

2 +

2)1 2

= 1

2 (

2 +

2)1 2 +

2

2 ln

+ (

2 +

2)1 2 .

19.

(

2

+

2

)

1 2

=

1

3 (

2

+

2

)

3 2

.

20.

(

2 +

2)3 2

= 1

4 (

2 +

2)3 2 + 3

8

2 (

2 +

2)1 2 + 3

8

4 ln

+ (

2 +

2)1 2

.

21.

1

(

2 +

2)1 2

= (

2 +

2)1 2 − ln

+ (

2 +

2)1 2

.

22.

2 +

2= ln

+ (

2 +

2)1 2 .

23.

2 +

2= (

2 +

2)1 2.

24.

(

2 +

2)−3 2

=

−2 (

2 +

2)−1 2.


2 −

2)1 2.

25.

(

2 −

2)1 2

= 1

2 (

2 −

2)1 2 −

2

2 ln

+ (

2 −

2)1 2

.

26.

(

2 −

2)1 2

= 1

3(

2 −

2)3 2.

27.

(

2 −

2)3 2

= 1

4 (

2 −

2)3 2 − 3

8

2 (

2 −

2)1 2 + 3

8

4 ln

+ (

2 −

2)1 2

.

28.

1

(

2 −

2)1 2

= (

2 −

2)1 2 − arccos

.

29.

2 −

2 = ln

+ (

2

−

2

)1 2

.

30.

2 −

2= (

2 −

2)1 2.

31.

(

2 −

2)−3 2

= −

−2 (

2 −

2)−1 2.


2 −

2)1 2.

32.

(

2 −

2)1 2

= 1

2 (

2 −

2)1 2 +

2

2 arcsin

.

33.

(

2 −

2)1 2

= −1

3(

2 −

2)3 2.

34.

(

2 −

2)3 2

= 1

4 (

2 −

2)3 2 + 3

8

2 (

2 −

2)1 2 + 3

8

4 arcsin

.

35.

1

(

2 −

2)1 2

= (

2 −

2)1 2 − ln

+ (

2 −

2)1 2

.

36.

2 −

2= arcsin

.

37.

2 −

2= −(

2 −

2)1 2.

38.

(

2 −

2)−3 2

=

−2 (

2 −

2)−1 2.

© 2003 by Chapman & Hall/CRC



S.3.2-6. Reduction formulas.

The parameters , , ,

, and below can assume arbitrary values, except for those at which

denominators vanish in successive applications of a formula. Notation:

=

+ .

39.

(

+

)

=

1

+ + 1

+1

+

−1

.

40.

(

+ )

= 1

( + 1)

−

+1

+1 + (

+ + + 1)

+1

.

41.

(

+ )

= 1

(

+ 1)

+1

+1 − (

+ + + 1)

+

.

42.

(

+ )

= 1

(

+ + 1)

− +1

+1 − (

− + 1)

−

.

S.3.3. Integrals Containing Exponential Functions

1.

=

1

.

2.

=

ln

.

3.

=

− 1

2

.

4.

2

=

2

− 2

2 +

2

3

.

5.

=

1

−

2

−1 + ( − 1)

3

−2 − + (−1) −1

!

+ (−1)

!

+1 ,

= 1, 2,

6.

( )

=

=0

(−1)

+1

( ), where

( ) is an arbitrary polynomial of degree .

7.

+

=

− 1

ln | +

|.

8.

+

−

=

1

arctan

if > 0,

1

2

−

ln

+

−

−

−

if

< 0.

9.

+

=

1

ln

+

−

+

+

if > 0,

2

−

arctan

+

−

if < 0.

S.3.4. Integrals Containing Hyperbolic Functions

S.3.4-1. Integrals containing cosh .

1.

cosh( + )

= 1

sinh( + ).

2.

cosh

= sinh − cosh .




3.

2 cosh

= (

2 + 2) sinh − 2 cosh .

4.

2 cosh

= (2 )!

=1

2

(2 )! sinh −

2 −1

(2 − 1)! cosh

.

5.

2 +1 cosh

= (2 + 1)!

=0

2 +1

(2 + 1)! sinh −

2

(2 )! cosh

.

6.

cosh

=

sinh −

−1 cosh + (

− 1)

−2 cosh

.

7.

cosh2

= 1

2 + 1

4 sinh 2 .

8.

cosh3

= sinh + 1

3 sinh3

.

9.

cosh2

=

2

22

+ 1

22 −1

−1

=0

2

sinh[2( − ) ]

2( − ) , = 1, 2,

10.

cosh2 +1

= 1

22

=0

2 +1

sinh[(2 − 2 + 1) ]

2 − 2 + 1 =

=0

sinh2 +1

2 + 1 , = 1, 2,

11.

cosh

= 1

sinh cosh

−1 +

− 1

cosh

−2

.

12.

cosh cosh

= 1

2 −

2 ( cosh

sinh

− cosh

sinh

).

13.

cosh

= 2

arctan

.

14.

cosh2

= sinh

2

− 1

1

cosh2 −1

+

−1

=1

2 ( − 1)( − 2) ( − )

(2

− 3)(2

− 5)

(2

− 2

− 1)

1

cosh2 −2 −1

,

= 1, 2,

15.

cosh2 +1

= sinh

2

1

cosh2

+

−1

=1

(2 − 1)(2 − 3) (2 − 2 + 1)

2 ( − 1)( − 2) ( − )

1

cosh2 −2

+ (2 − 1)!!

(2 )!! arctan sinh , = 1, 2,

16.

+ cosh

=

− sign

2 −

2arcsin

+ cosh

+ cosh

if

2 <

2,

1

2 −

2ln

+ +

2 −

2 tanh(

2)

+ −

2 −

2 tanh(

2)if

2 >

2.

S.3.4-2. Integrals containing sinh .

17.

sinh( + )

= 1

cosh( + ).

18.

sinh

= cosh − sinh .

19.

2 sinh

= (

2 + 2) cosh − 2 sinh .

20.

2 sinh

= (2 )!

=0

2

(2 )! cosh −

=1

2 −1

(2 − 1)! sinh

.




21.

2 +1 sinh

= (2 + 1)!

=0

2 +1

(2 + 1)! cosh −

2

(2 )! sinh

.

22.

sinh

=

cosh −

−1 sinh + (

− 1)

−2 sinh

.

23.

sinh2

= − 1

2 + 1

4 sinh 2 .

24.

sinh3

= − cosh + 1

3 cosh3

.

25.

sinh2

= (−1)

2

22

+ 1

22 −1

−1

=0

(−1)

2

sinh[2( − ) ]

2( − ) , = 1, 2,

26.

sinh2 +1

= 1

22

=0

(−1)

2 +1

cosh[(2 − 2 + 1) ]

2 − 2 + 1 =

=0

(−1) +

cosh2 +1

2 + 1 ,

= 1, 2,

27.

sinh

=

1

sinh

−1

cosh

−

− 1

sinh

−2

.

28.

sinh sinh

= 1

2 −

2

cosh sinh

− cosh

sinh

.

29.

sinh

= 1

ln

tanh

2

.

30.

sinh2

= cosh

2 − 1 −

1

sinh2 −1

+

−1

=1

(−1) −1 2 ( − 1)( − 2)

( − )

(2 − 3)(2 − 5) (2 − 2 − 1)

1

sinh2 −2 −1

, = 1, 2,

31.

sinh2 +1

=

cosh

2

−

1

sinh2

+

−1

=1(−1)

−1 (2 −1)(2 −3)

(2 −2 +1)

2 ( −1)( −2) ( − )

1

sinh2 −2

+ (−1)

(2 − 1)!!

(2 )!! ln tanh

2 , = 1, 2,

32.

+ sinh

= 1

2 +

2ln

tanh(

2) − +

2 +

2

tanh(

2) − −

2 +

2.

33.

+ sinh

+ sinh

=

+

−

2 +

2ln

tanh(

2) − +

2 +

2

tanh(

2) − −

2 +

2.

S.3.4-3. Integrals containing tanh or coth .

34.

tanh

= ln cosh .

35.

tanh2

= − tanh .

36.

tanh3

= − 1

2 tanh2

+ ln cosh .

37.

tanh2

= −

=1

tanh2 −2 +1

2 − 2 + 1 , = 1, 2,

38.

tanh2 +1

= ln cosh −

=1

(−1)

2 cosh2

= ln cosh −

=1

tanh2 −2 +2

2 − 2 + 2 , = 1, 2,




39.

tanh

= − 1

− 1 tanh

−1 +

tanh

−2

.

40.

coth

= ln |sinh |.

41. coth2

= − coth .

42.

coth3

= − 1

2 coth2

+ ln |sinh |.

43.

coth2

= −

=1

coth2 −2 +1

2 − 2 + 1 , = 1, 2,

44.

coth2 +1

= ln |sinh | −

=1

2 sinh2

= ln |sinh |−

=1

coth2 −2 +2

2 − 2 + 2 , = 1, 2,

45.

coth

= − 1

− 1 coth

−1 +

coth

−2

.

S.3.5. Integrals Containing Logarithmic Functions

1.

ln

= ln − .

2.

ln

= 1

2

2 ln − 1

4

2.

3.

ln

=

1

+ 1

+1 ln −

1

( + 1)2

+1 if ≠ −1,

1

2 ln2

if = −1.

4.

(ln )2

= (ln )2 − 2 ln + 2 .

5.

(ln )2

= 1

2

2(ln )2 − 1

2

2 ln + 1

4

2.

6.

(ln )2

=

+1

+ 1(ln )2 −

2

+1

( + 1)2

ln + 2

+1

( + 1)3

if ≠ −1,

1

3 ln3

if = −1.

7.

(ln )

=

+ 1

=0

(−1) ( + 1) ( − + 1)(ln ) − , = 1, 2,

8.

(ln )

= (ln ) −

(ln )

−1

, ≠ −1.

9.

(ln )

=

+1

+ 1

=0

(−1)

( + 1) +1 (

+ 1)

(

− + 1)(ln ) − , ,

= 1, 2,

10.

(ln )

= 1

+ 1

+1(ln ) −

+ 1

(ln )

−1

, , ≠ −1.

11.

ln( + )

= 1

( + ) ln(

+ ) − .

12.

ln( + )

= 1

2

2 −

2

2

ln( + ) −

1

2

2

2 −

.

13.

2 ln( + )

= 1

3

3 −

3

3

ln( + ) −

1

3

3

3 −

2

2

+

2

2

.




14.

ln

( + )2

= − ln

( + )

+ 1

ln

+

.

15.

ln

( + )3

= − ln

2 ( + )2

+ 1

2

( + )

+ 1

2

2

ln

+

.

16.

ln

+

=

2

(ln − 2)

+ +

ln

+

+

+ −

if > 0,

2

(ln − 2)

+ + 2 − arctan

+

−

if < 0.

17.

ln(

2 +

2)

= ln(

2 +

2) − 2 + 2 arctan(

).

18.

ln(

2 +

2)

= 1

2

(

2 +

2) ln(

2 +

2) −

2 .

19.

2 ln(

2 +

2)

= 1

3

3 ln(

2 +

2) − 2

3

3 + 2

2 − 2

3 arctan(

) .

S.3.6. Integrals Containing Trigonometric Functions

S.3.6-1. Integrals containing cos ( = 1, 2,

).

1.

cos( + )

= 1

sin( + ).

2.

cos

= cos + sin .

3.

2 cos

= 2 cos + (

2 − 2) sin .

4.

2

cos

= (2 )!

=0

(−1)

2 −2

(2 − 2 )! sin +

−1

=0

(−1)

2 −2 −1

(2 − 2 − 1)! cos .

5.

2 +1 cos

= (2 + 1)!

=0

(−1)

2 −2 +1

(2 − 2 + 1)! sin +

2 −2

(2 − 2 )! cos

.

6.

cos

=

sin +

−1 cos − ( − 1)

−2 cos

.

7.

cos2

= 1

2 + 1

4 sin 2 .

8.

cos3

= sin − 1

3 sin3

.

9.

cos2

= 122

2

+ 122 −1

−1

=0

2

sin[(2 − 2 ) ]2 − 2

.

10.

cos2 +1

= 1

22

=0

2 +1

sin[(2 − 2 + 1) ]

2 − 2 + 1 .

11.

cos

= ln

tan

2 +

4

.

12.

cos2

= tan .

13.

cos3

= sin

2 cos2

+ 1

2 ln

tan

2 +

4

.




14.

cos

= sin

( − 1) cos −1

+ − 2

− 1

cos −2

, > 1.

15.

cos2

=

−1

=0

(2 − 2)(2 − 4) (2 − 2 + 2)

(2 − 1)(2 − 3) (2 − 2 + 3)

(2 − 2 ) sin − cos

(2 − 2 + 1)(2 − 2 )cos2 −2 +1

+ 2

−1( − 1)!

(2 − 1)!!

tan + ln |cos | .

16.

cos cos

=sin ( − )

2( − ) +

sin ( + )

2( + ) , ≠

.

17.

+ cos

=

2

2 −

2arctan

( − ) tan(

2)

2 −

2if

2 >

2,

1

2 −

2ln

2 −

2 + ( − ) tan(

2)

2 −

2 − ( − ) tan(

2)

if

2 >

2.

18.

( + cos )2 =

sin

(

2 −

2)( + cos ) −

2 −

2

+ cos

.

19.

2 +

2 cos2

= 1

2 +

2arctan

tan

2 +

2.

20.

2 −

2 cos2

=

1

2 −

2arctan

tan

2 −

2if

2 >

2,

1

2

2 −

2ln

2 −

2 − tan

2 −

2 + tan

if

2 >

2.

21.

cos

=

2 +

2 sin

+

2 +

2 cos

.

22.

cos2

=

2 + 4

cos2 + 2 sin cos +

2

.

23.

cos

=

cos −1

2 +

2 ( cos + sin ) +

( − 1)

2 +

2

cos −2

.

S.3.6-2. Integrals containing sin ( = 1, 2, ).

24.

sin( + )

= −1

cos( + ).

25.

sin

= sin − cos .

26.

2 sin

= 2 sin − (

2 − 2)cos .

27.

3 sin

= (3

2 − 6) sin − (

3 − 6 ) cos .

28.

2 sin

= (2 )!

=0

(−1) +1

2 −2

(2 − 2 )! cos +

−1

=0

(−1)

2 −2 −1

(2 − 2 − 1)! sin

.

29.

2 +1 sin

= (2 + 1)!

=0

(−1) +1

2 −2 +1

(2 − 2 + 1)! cos + (−1)

2 −2

(2 − 2 )! sin

.

30.

sin

= −

cos +

−1 sin − (

− 1)

−2 sin

.

31.

sin2

= 1

2 − 1

4 sin 2 .






50.

sin

=

2 +

2 sin

−

2 +

2 cos

.

51.

sin2

=

2 + 4

sin2 − 2 sin cos +

2

.

52.

sin

=

sin −1

2 +

2 ( sin − cos ) +

(

− 1)

2 +

2

sin −2

.

S.3.6-3. Integrals containing sin and cos .

53.

sin cos

= −cos[( + ) ]

2( + ) −

cos ( − )

2( − ) , ≠

.

54.

2 cos2

+

2 sin2

= 1

arctan

tan

.

55.

2 cos2

−

2 sin2

= 1

2

ln

tan +

tan −

.

56.

cos2

sin2

=

+ −1

=0

+ −1

tan2 −2 +1

2 − 2

+ 1 , ,

= 1, 2,

57.

cos2 +1 sin2 +1

=

+

ln |tan | +

+

=0

+

tan2 −2

2 − 2

, ,

= 1, 2,

S.3.6-4. Reduction formulas.

The parameters and

below can assume any values, except for those at which the denominators

on the right-hand side vanish.

58.

sin

cos

= − sin

−1 cos

+1

+

+

− 1

+

sin

−2 cos

.

59.

sin

cos

= sin

+1 cos

−1

+

+ − 1

+

sin

cos

−2

.

60.

sin

cos

= sin

−1 cos

−1

+

sin2 −

− 1

+

− 2

+ (

− 1)(

− 1)

( + )( + − 2)

sin

−2 cos

−2

.

61.

sin

cos

= sin

+1 cos

+1

+ 1

+ +

+ 2

+ 1

sin

+2 cos

.

62.

sin

cos

= − sin

+1

cos

+1

+ 1

+

+

+ 2

+ 1

sin

cos

+2

.

63.

sin

cos

= −sin

−1 cos

+1

+ 1

+ − 1

+ 1

sin

−2 cos

+2

.

64.

sin

cos

= sin

+1 cos

−1

+ 1 +

− 1

+ 1

sin

+2 cos

−2

.

S.3.6-5. Integrals containing tan and cot .

65.

tan

= − ln |cos |.






10.

arctan

= 1

2(

2 +

2) arctan

−

2 .

11.

2 arctan

=

3

3 arctan

−

2

6 +

3

6 ln(

2 +

2).

12.

arccot

= arccot

+

2 ln(

2 +

2).

13.

arccot

= 1

2(

2 +

2) arccot

+

2 .

14.

2 arccot

=

3

3 arccot

+

2

6 −

3

6 ln(

2 +

2).

References for Subsection S.3.3: H. B. Dwight (1961), I. S. Gradshteyn and I. M. Ryzhik (1980), A. P. Prudnikov,Yu. A. Brychkov, and O. I. Marichev (1986, 1988).




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