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.
S.3.2-4. Integrals containing (
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.
S.3.2-5. Integrals containing (
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.
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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 .
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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
.
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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,
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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
.
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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
.
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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 .
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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 |.
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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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