Tabella Integrali Fondamentali analisi matematica 1 PDF

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Tabella Integrali

INTEGRALI FONDAMENTALI

A cura di Gaetano Cioppa

2

© Copyright 2003 by Cioppa Gaetano Tutti i diritti sono riservati La presente dispensa può essere copiata, fotocopiata, riprodotta, a patto che non venga alterata ed utilizzata a scopo di lucro, la proprietà del documento rimane di Cioppa Gaetano. Per ulteriori informazioni si prega di contattare l’autore all’indirizzo: [email protected]

3

Dedicato a Me

“Gli ideali che hanno illuminato il mio cammino e che spesso mi hanno dato nuovo coraggio per affrontare la vita con allegria sono stati la gentilezza la bellezza e la verità.” (Albert Einstein)

4

5

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

dx == x x λ +1 x λ dx = λ +1 n

dx =

x

n n +1

n

x

n +1

1 1  x dx = arctg   2 a x +a a 2

1 1 x−a dx = log 2 2a x+a x −a 2

1 1 a+x dx = log 2 2a a−x a −x 2

1 dx = log x + a 2 + x 2 x2 + a2  x x dx = arcsen   = − arccos   a  a a2 − x2 1

1 x2 − a2 1

dx

+ b

ax

x 2− a2

dx = log x +

=

2 a

a x + b

1 1 log a x + b dx = a a x +b 1 1 + (a x + b)

2

dx =

1 arctg (a x + b) a

1 a x +b 1 dx = − log b x x(ax + b) 1 x

x2 + a2

dx = −

1 a

a+ log

x 2+ a 2 x

6

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

1 x2

x2 + a2

1 x( x2 + a2 ) 1 ax− 1

x

1  +x2   1   1 x  x 2 + 1   

a2 x

 x2 log   2 2 2a2  x +a 1

dx =

x

(

1

cx

dx = log

x 2 ++1

x 3+ x 2 1 x6+ x4

2

)

dx = −

1 arctg x 2

+

2  1 + x 2   

x +1 dx = log x 2 x2 1 +x 1

   

dx == 2 arctg a x −−1

dx =

2

x2 + a2

dx = −

−

2x + 1

(

)

x 1 ++ x 2

x ++ 1 1 + log x x

dx == arctg x ++

1 1 −− x 3x 3

(

)

(

)

 x−1 2 1 1 log  dx =  2+ x + 1 6 x 3− 1  x

  −  

 2x + 1 arctg   3 3 

1

   

    x +1 2  1 1 1  2x − 1  log  dx = arctg  +   2 − x +1  6  3  3 x 3+ 1   x      1 x −1 1 dx = log  4 arctg x  − 2 x +1   x4 − 1       x 2  x2 + x 2 + 1  2 2  1  dx = log  arctg  +   4 4 2  x2 − x 2 + 1  x4 + 1   1− x    x −1 1 dx = log   x  x2− x 1 1 log dx = 2 x2− 1

   + 

  k + c 

   

x−1 == −− arccotgh x x+1

7

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

 dx = log  x +  x2−1 1

1 1+ x 2

 x 2 − 1  = settcosh x 

dx = arctg x = − arcctg x

1 1+ x 2

  dx = log  x + x 2 + 1  = settsenh x  

1 x +1 1 = setttgh x = arctgh x dx = log 2 2 x −1 1− x 1

dx = arcsen x = − arccos x

1− x2 x

dx =

a x +b

2 ( a x + b) 3a2

a x +b

x b x dx = − 2 log a x + b a a a x +b bp− aq a x +b a log p x + q + c dx = x + p p x +q p2

b 1 x dx = 2 + 2 log a x + b 2 a ( a x + b) a (ax + b) 1 x log  x 2 + a 2  dx = 2   2 2 x +a

(

1 x dx = log x 2 − a 2 2 2 x −a 2

)

1 x dx = − log a 2 − x 2 2 2 a −x 2

x x −a 2

2

x x +a 2

2

x a −x 2

2

dx = x 2 − a 2 dx =

x2 +a2

dx = −

a2 − x2

8

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

∫( ∫( ∫( ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

x a −x 2

x a2 + x

)

dx = −

)

dx = −

2 n

x x2 − a

)

dx =

2 n

2 n

(

1

2 (n − 1) a 2 − x 2

(

)

n −1

1

2 (n − 1) a 2 + x 2

(

)

1

2 (n − 1) x 2 − a 2

n −1

)

n −1

x x2 dx = x − a arctg 2 2 a x +a x2 a −x 2

2

x2 x2 + a2 x2 x2 − a2

dx =

a2 x  1 arcsen   − 2 a  2

dx =

x 2

x 2 +a 2 −

a2 log x + x 2 + a 2 2

dx =

x 2

x2 −a2 +

a2 log x + x 2 − a 2 2

a 2 − x2

(

)

(

)

x 2 + a 2 dx =

x 2

x2 +a2 +

a2 log x + x 2 + a 2 2

x 2 − a 2 dx =

x 2

x2 −a2 −

a2 log x + x 2 − a 2 2

a 2 − x 2 dx =

a2 x x arcsen + 2 a 2

a 2 − x 2 dx = −

x

x +a

2

x

x −a

2

x2

(a 2 − x 2)

3

x

x2

a2 −x2

3

(x 2 + a 2)

3

2

dx =

3

(x 2 − a 2)

3

2

dx =

a 2 − x 2 dx =

3 x 4

2 2 4  2 a2  a 2 − x 2 + a arcsen 2x − a x −  2  16 a2 

(

)

1 2 2 x +a x x 2 + a 2 dx = 4

2 a 4 log  2 + a  + x x x 2 + a2 −  2  16 

 x2 + a2  

9

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

x2

(

)

2  a4 a  log  x 2 −  + x x2−a2 − 2 16 

1 2 x − a2 x 4

x 2 − a 2 dx =

a 2− x 2 x

dx =

a 2 − x 2 − a log

a2 − x2 x2

dx = −

1 x

2 3a

a x + b dx = 1+ x x 1+ x 2

x

dx = 2

dx =

dx =

x

dx =

1 − x4

x a

(a x + b ) 3

( a x + b)3

( x + 1 + log

)

x + 1 −1 x

 1 + x −1  1  − log   2  1+ x + 1

x2 − 1 x

a2− x2 x

a +

a 2 − x 2 − arcsen

2 (3 a x − 2b ) 15 a 2

x a x + b dx =

 x2 −a2  

x 2 − 1 − arctg

2

1+ x x

x 2 −1

1 arcsen x 2 2

1 1  x +1  x2 dx = log   − arctg x 4 4 2 1− x  x −1 

x3

dx =

 1− 2   x   

1− x 2

3

 x x dx = arctg  1− x  1− x x− 1 dx = x − 2 log ( x +1 x+ 1 1− x dx = arcsen x + 1+ x x

1− x dx = arctg 1+ x

−−

1 −− x 2

  −  

x − x2

3

)

1- x2

1− x ( x − 2) + 1+ x 2

1 − x2

10

74.

75.

76.

77.

78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

(ax + b )

n

dx =

(a x + b )n +1 a (n +1) x (a x + b) n a(n + 1)

x ( ax + b) dx = n

+1

−

(a x + b) n+ 2 a 2 (n + 1)(n + 2)

ax x a dx = log a ax

e e ax dx = a xa

dx =

x

x ax ax − log a log 2 a

2 x x 2 ax x 2 a x dx = x a − 2 x a + log a log 2 a log 3 a

xe x e ax dx = a x2 e

x x e

2

x

e ax a2

− ax

2 ax dx = x e a

x e− 2

3

ax

−

dx = − x e− 2

2 ea x 2 x ea x + a2 a3 x

−

2

dx = x e 2

x2 2

e ax cos (b x ) dx = e ax sen(b x ) dx =

x n e λ x dx =

x e− 2

x

−

e− 2 2

x

2

− 2e

x 2

e a x [a cos (b x )+ b sen (b x )] a 2 +b2 e a x [a sen (b x ) − b cos (b x )] a 2 + b2

x n eλ x n − λ λ

∫

x n −1 e λ x dx

[

]

x n e x dx = e x x n − nx n− 1 + n(n − 1)x n− 2 − ..... + (− 1) n n !

e arcsen x dx =

(

1 arcsen x e x + 1− x 2 2

) 11

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

sen x cos x e sen x dx = 2

x e −x dx = −

e −x 2

∫

sen ( 2 x) sen x e dx = sen x e sen x − e sen x 2

2

(

1 dx = x − log 1 + e x 1 + ex

)

ex − 1 dx = log e x + e −x + 2 x e +1

(

e 2x 1+e

dx = x

(

2 x e −2 3

)

)

1 + ex

sen x + cos x senx dx = − x 2 ex e  e2x 1 1 dx = log  2 x 2x 2 1+e e + e x − 1 dx = 2 cosh (ax ) dx =

  1 

e x − 1 − 2 arctg e x − 1

senh (a x ) a

x cosh( ax) dx =

x cosh( ax) senh (ax ) − a a2

( )

2 1 dx = arctg e ax a cosh ( a x) 1 dx = tgh x cosh 2 x senh (ax ) dx =

cosh (a x ) a

x senh (a x ) dx =

x senh (a x ) cosh (a x ) − a a2

1 1  a x dx = log tgh   a senh ( a x)  2 

12

104.

105.

106.

107.

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109.

110.

111.

112.

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114.

115.

116.

117.

118.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

1 dx = − cotgh x senh2 x sen(a x ) ⋅ cos(a x ) dx =

sen 2 (a x ) 2a

sen(a x) ⋅ cos(b x) dx = −

cos (a + b) x cos (a −b ) x − 2 (a + b ) 2 (a − b )

sen(a x)⋅ sen(b x ) dx =

sen (a − b ) x sen (a + b ) x − 2 ( a − b) 2 (a + b )

cos (a x ) ⋅ cos (b x ) dx =

sen (a − b) x sen (a + b ) x + 2 (a − b ) 2 (a + b )

senx cosx dx =

sen 2 x 2

sen 2 x cosx dx =

sen 3 x 3 cos3 x 3

cos 2 x sen x dx = −

n +1

x sen sen x cosx dx = n+ 1 n

cos n x senx dx = −

cosn +1 x n +1

1 dx = log tg x senx cosx 1 dx = tg x − cotg x sen x cos2 x 2

sen 2 x − cos 2 x dx = tg x + cotg x sen 2 x cos 2 x 1 senx dx = 2 cos x cos x 1 cos x dx = − 2 sen x sen x

13

119.

120.

121.

122.

123.

124.

125.

126.

127.

128.

129.

130.

131.

132.

∫ ∫ ∫ ∫ ∫ ∫

sen (log x )dx = senx cosx 1 + cos x 2

x [sen (log x ) − cos (log x )] 2

dx = − 1 + cos 2 x

1 senx dx = 3 2cos2 x cos x sen 3 x 1 sen x − 1 sen 4 x dx = − − sen x − log 3 2 sen x + 1 cosx 1 1  2 tg x +1  sen 2 x dx = − log ( 1+ senx cosx ) + arctg   2 senx cosx + 1 3 3  

∫

∫

cosx 1 cotg x dx = dx = dx = senx tg x 1 1 log sen x + cos x + sen 2x + arcsen (sen x − cos x ) = 2 2

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

senx cos x

(

)

dx = −2 cos x x2

( x cos x − sen x)

2

dx =

x sen x + cos x x cos x − sen x

sen x − sen 3 x 1 x dx = sen x cos x − cos x − 1 + sen x 2 2

sen 2 x dx = x + sen x 1 − cos x

cos 2x dx = senx + cosx senx + cosx

1 + sen2x dx = tg x − 2 log cos x cos 2 x  1 + senx x x dx = tg + log 1 + tg 2    1 + cosx 2  2  1 + senx x x e dx = e x tg 1 + cosx 2

14

133.

134.

135.

136.

137.

138.

139.

140.

141.

142.

143.

144.

∫ ∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

x 1 − cosx dx = 2 tg − x 2 1 + cosx 1 − senx dx = log ( x + cos x x + cosx

)

 1 + sen x  x + cos x dx = x   1 − sen x  cos x  cosx dx = arctg (sen x ) 1 + sen 2 x 1 1  ax π  dx = log tg  ±  sen( a x) ± cos( a x) a 2  2 8 x 1 cos( a x) dx = ± + log sen (a x ) ± cos (a x ) 2 2a sen( a x) ± cos( a x) x 1 sen( a x) dx = m log sen ( a x ) ± cos ( a x ) 2 2a sen( a x) ± cos( a x)

   1 se a b 〉  ) 1  dx =  + a b cos x   2 ) se b 〉 a   

1 dx = a cosx + b senx

 a +b x arctg  ⋅ tg   2 a −b  a-b 1

2

2

 x b+a  tg + 1 b−a log  2  2 2 x b+a b −a  tg ⋅ 2 b−a 

  a tg 1 log   a tg a2 +b2  

x − b + a 2 + b2 2 x − b − a 2 + b2 2

     

     

1 tg( ax) dx = − log cos (a x ) a tg 2 x dx = tg x − x

tg 4 x dx =

1 3 tg x − tg x + x 3

15

145.

146.

147.

148.

149.

150.

151.

152.

153.

154.

155.

156.

157.

158.

∫ ∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫ ∫

tg n x dx =

∫

1 tg (n -1 ) x = − dx cotg n x (n − 1)

tg (n +1 ) (a x ) tg n x dx = 1 + sen 2 x a (n +1)

(

1 tgx dx = log 2 tg 2x +1 2 4 1 + sen x

∫

tg n −2 x dx

)

tg 3 x + tg x dx = tg x − 4 log 4 + tg x tg x + 4 cotg(a x ) dx =

1 log sen (a x) a

cotg 2 x dx = − cotg x − x

cotg n x dx =

∫

1 cotg n -1x = − − dx tg n x (n −1)

∫

cotg (n - 2 )x dx

  x 1) per arccosh  a  〉 0    x  arccosh  dx =  a  x  2 ) per arccosh  a  〈 0   

x x arccosh   − x 2 − a 2 a  x  x arccosh   + x 2 − a 2 a 

x x arcsenh   dx = x arcsenh   − x 2 + a 2 a a  tgh(a x ) dx =

1 log cosh ( a x ) a

x a  x 2 2 arctgh   dx = x arctgh   + log a − x a  2 a  x   x  a arccotgh   dx = x arccotgh   + log x 2 − a 2 a a     2 cotgh (a x ) dx = cos ( a x) dx =

1 log senh ( a x a

1 sen ( a x a

)

)

16

159.

160.

161.

162.

163.

164.

165.

166.

167.

168.

169.

170.

171.

172.

173.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

x cos (a x ) dx =

1 ( ) + x sen ( a x 2 cos a x a a

)

x 2 cos x dx = x 2senx + 2 x cosx − 2 senx

( x ) dx = 2

cos

x sen x + 2 cos x

cos 2 x dx =

x 1 1 x + sen ( 2 x ) = sen x cos x + 2 4 2 2

cos 3 x dx =

1 2 1 cos 2 x sen x + sen x = sen x − sen3 x 3 3 3

1 1  ax π  dx = log tg  +  a 4  cos (a x )  2 1 a x  1 dx = tg   a  2  1 + cos (a x ) 1 1  a x dx = − cotg   a 1 − cos (a x )  2  1 dx = tg x cos2 x 1 1 dx = tg x + tg 3 x 4 3 cos x sen x n− 2 1 dx = ( n −1) + n (n −1) cos x n − 1 cos x cos n x dx =

1 n −1 cos n −1 x sen x + n n

∫ ∫

1 cos n− 2 x

dx

cosn −2 x dx

x 2 x a x   a x dx = tg   + 2 log cos   a 1 + cos (a x )  2   2  a x 2 x  a x a x  dx = − cotg   + 2 log sen   a 1 − cos (a x )  2  a  2  1  a x cos (a x ) dx = x − tg   a  2  1 + cos (a x )

17

174.

175.

176.

177.

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179.

180.

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182.

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184.

185.

186.

187.

188.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

1 cos( a x)  a x dx = − x − cotg   a 1 − cos (a x )  2  x dx = x tg x + log cos x cos2 x

( )

1 x dx = tg x 2 2 2 2 cos x

x m cos x dx = x m sen x − m

cos (logx ) dx =

∫

x m −1sen x dx

x [sen (log x) + cos (log x)] 2

sen (a x ) dx = −

1 cos (a x ) a

x sen (a x )dx =

1 x sen (a x ) − cos (a x ) 2 a a

x 2 senx dx = − x 2 cosx + 2 x senx + 2 cosx sen 2 x dx =

x 1 x 1 − sen2x = − senx cosx 2 4 2 2

sen 3 x dx = − cos x +

1 cos 3 x 3

1 1  a x dx = log tg   a sen (ax )  2  1 a x π 1 dx = tg  −  a  2 4 1 + sen( a x)

2 1 dx = − x 1 + senx 1 + tg 2 1 a x π 1 dx = tg  +  a  2 4 1 − sen( a x)

2 1 dx = x 1 − senx 1 − tg 2 18

189.

190.

191.

192.

193.

194.

195.

196.

197.

198.

199.

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

x a x π 2  x  a x π  dx = tg  −  −  + 2 log cos  a  2 4 a 1 + sen( a x)  2 4   x x π a x 2 π ax dx = cotg  −  + 2 log sen  −  2  a a 1 − sen( a x) 4 4 2  1 π ax sen( a x) dx = x + tg  −  a 4 2  1 + sen( a x) 1 π a x sen( a x) dx = −x + tg  +  a 4 2  1 − sen( a x) cos x n−2 1 dx = − + n n −1 (n − 1)...