Tabella Integrali

2

INTEGRALI

FONDAMENTALI

A cura di Gaetano Cioppa

3

© 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: ci.tano@inwind.it

4

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)

5

6

1. ∫ = x dx

2. 1λ

x 1λ

+

+=∫ dxx λ

3. n 1 n x 1 n

n +

+=∫ dx x n

4. ax arctg

a1 1

22

=

+∫ dxax

5. a x a x log

a 21 1

22 +−

=−∫ dx

ax

6. x a x a log

a 21 1

22 −+

=−∫ dx

xa

7. x2a2 x log ++=+∫ dx

a 2x 21

8.

−=

=

−∫ ax arccos

axarcsen dx

x 2 a 21

9. a2 x2 x log −+=−∫ dx

a 2 x 21

10. ∫ +=+

b x a a2 dx

ba x 1

11. ∫ +=+

b xa log a1

dx

bxa1

12. b) x(a arctg a1 +=

++∫ dx b)(a x1

12

13. ∫ +−=

+

xb xa log

b1

dx

b )x x ( a1

14. x

a 2x 2a log

a1

++

−=+∫ dx

a 2 x 2x

1

7

15. x2a

a 2 x 2

+−=

+∫ dx a 2 x 2x 2

1

16.

+=

+∫ a 2x 2x 2

log 2a 2

1 dx

)a 2 x 2x (

1

17. ∫ −=−

1 xa arctg 2 dx1a xx

1

18. xarctg 21

2 x 1 2

x

221

+

+

=

+∫ dx

x

1

19. ∫ +=

+

1x 2

cx log dx

1x 2x

1

20. ( ) ( )∫ +

+−

+=

+ 2x1x

12x

2

x1x

log dx2x1x 2

1

21. ∫ ++−=

+ x1x

log x1

dxx 2x 3

1

22. ∫ −+=+ 33x

1

x1

xarctg dxx 4x 6

1

23. ( )

+−

++

−=

−∫ 3

12x arctg

3

1

1xx 2

21x log

61

dx1x 3

1

24. ( )

−+

+−

+=

+∫ 3

12x arctg

3

1

1xx 2

21x log

61

dx1x 3

1

25. xarctg 21

4 1x1x

log −

+−

=−∫ dx

1x 41

26. c kx 21

2x arctg

42

12xx 212xx 2

log 42

+

+

−+

+−

++=

+∫ dx1x 4

1

27.

−=

−∫ x1x

log dx xx 2

1

28. ∫ −=+−

=−

arccotgh x 1x1x

log 21

dx1x 2

1

8

29. ∫ =

−+=

−settcosh x 1x 2x log dx

1x 2

1

30. ∫ −==+

xarcctg x arctg dxx 21

1

31. settsenh x 1x 2x log =

++=

+∫ dx x 21

1

32. ∫ ==−+

=−

arctgh x setttgh x 1x1x log

21 dx

x11

2

33. xarccos arcsen x −==−∫ dx

x 21

1

34. ( )∫ ++=+

b xa b xa a 32

2 dxbxa

x

35. ∫ +−=+

b xa log ab

ax

2 dxbxa

x

36. ∫ ++−

+=++ c q xp log

pq ap b x

pa

2 dxqxpbxa

37. ( ) ( )∫ ++

+=

+ b xa log

a1

b xaab 22 dx

baxx

2

38. ∫

+=

+2a2x log

21

dx2a2x

x

39. ( )∫ −=−

22 a x log 21 dx

axx

22

40. ∫ −−=−

xa log 21 22 dx

xax

22

41. ∫ −=−

22 ax dxax

x22

42. ∫ +=+

22 ax dxax

x22

43. ∫ −−=−

xa 22 dx xa

x22

9

44. ( ) ( ) ( )∫ −

−−=

−1n 22 xa 1n 2

1 dxxa

xn22

45. ( ) ( ) ( )∫ −

+−−=

+1n 22 xa 1n 2

1 dxxa

xn22

46. ( ) ( ) ( )∫ −

−−−=

−1n 22 ax 1n 2

1 dxax

xn22

47. axarctg a x ∫ −=

+ dx

axx

22

2

48. ∫ −−

=

−22

2

xa 21

axarcsen

2a dx

xax

22

2

49. ( )∫ ++−+=+

222

22 axx log 2

a ax 2x dx

axx

22

2

50. ( )∫ −++−=−

222

22 axx log 2

a ax 2x dx

axx

22

2

51. ∫ ++++=+ ax x log 2

a ax 2x 22

222 dx ax 22

52. ∫ −+−−=− ax x log 2

a ax 2x 22

222 dx ax 22

53. ∫ −+=− 222

xa 2x

axarcsen

2a dx xa 22

54. ( )∫ −

−=−3

x2a2

3

dx x 2a 2x

55. ( )∫ +

=+3

a2x2

3

dx axx 22

56. ( )∫ −

=−3

a2x2

3

dx axx 22

57. a2

a2xarcsen 16a4

x2a2 2

a2 x

4x

222 −

+−

−=−∫ dxx 2a 2 x 2

58. ( )

++

+−++=+∫ a2x2x

2ax2 log

16a4

ax2 x ax2 41

222 dxa 2x 2 x 2

10

59. ( )

−+

−−−−=−∫ a2x2x

2ax2 log

16a4

ax2 x ax2 41

222 dx a 2x 2 x 2

60. x

x2a2 a log a x2a2 −+−−=

−∫ dxx

x2a 2

61. axarcsen x2a2

x1

−−−=−∫ dx

x2x2a2

62. ( ) ( )∫ +−

=+ b xa a215

2b xa 3 2 3 dxb xax

63. ( )∫ +=+ b xa a 3

2 3 dxb xa

64. ( )∫ −+++=

+x

1 1 x log 1 x 2 2

dxx

x 1

65. x

x 1 1 x 11x 1 log

21 +

−

++−+

=+∫ dxx

x12

66. ∫ −−−=− 1x2 arctg 1x2 dx

x1x 2

67. ∫ =−

2arcsen x 21 dx

x41

x

68. ∫ −

−+

=−

xarctg 21

1x1x log

41 dx

x1x

4

2

69. ∫ −−

−

=−

x 21 3

3x 21

dxx 21

x 3

70. ∫ −−

−=

−xx

x1x arctg 2 dx

x1x

71. ( )∫ +−=+− 1 x log 2 x dx

1x1x

72. ∫ +=+− x2-1 arcsen x dx

x1x1

73. ( )∫ −−

++−

=+−

x1 2

2x x1x1 arctg 2 dx

x1x1x

11

74. ( ) ( )( )∫ ++

=++

1nab xa

1n

dx bax n

75. ( ) ( )( )

( )( )( )∫ ++

+−

++

=+++

2n1nab xa

1nab xax 2

2n1n

dxbaxx n

76. ∫ =a log

a x dxa x

77. a

e ax

=∫ dxeax

78. alog2

a x

a logax x

−=∫ dx x a x

79. alog3

a2 alog2

a x 2 x

a loga x x

x2

+−=∫ dx ax x2

80. 2

ax xa

ae

aex −=∫ dxx eax

81. a3e xa 2

a2e x 2 xa

ae x

2 ax

+−=∫ dx ex ax2

82. 2

ee xe x x 2

x 2x 2 −

−−− −−−=∫ dx ex x2

83. ∫ −= 22x

2x

e 2 e x 22

dxex

x23

2

84. ( ) ( ) ( )[ ]22

xa

ba xbsen b xb cos ae

++

=∫ dxxbcose ax

85. ( ) ( ) ( )[ ]22

xa

ba xb cos b xbsen ae

+−

=∫ dx xbsene ax

86. dx e xλn

λe x xλ1n

xλn ∫∫ −−= dx ex xλn

87. ( ) ( )[ ]!n 1.....x1nnnxx e n2n1nnx −+−−+−= −−∫ dxex xn

88. ( )2arcsen x x1xe 21 −+=∫ dx e arcsen x

12

89. ( ) sen xsen xsen x e esen x dx e 2

x2sen −== ∫∫ dx x esen x cos sen x

90. 2

e 2x−

− −=∫ dxx e2x

91. ( )∫ +−=+

xe1 log x dxe1

1x

92. ( )∫ ++=+− − 2 e elog xx dx

1e1e

x

x

93. ( )∫ +−=+

e1 2e 32 xx

dxe1

ex

x2

94. ∫ +−= xe 2

xcos sen x dx e

senxx

95. ∫

+

=+ 1 e

e log 21 x2

x2

dxe11

x2

96. ∫ −−−=− 1 e arctg 2 1e 2 xx dx 1 e x

97. ( ) ( )∫ =a

xasenh dxaxcosh

98. ( ) ( ) ( )∫ −= 2aaxcosh axsenh

ax dx axx cosh

99. ( ) ( )∫ = axe arctg a2

dx

xacosh1

100. ∫ = tgh x dx xcosh

12

101. ( ) ( )∫ =a

xacosh dxaxsenh

102. ( ) ( ) ( )∫ −= 2a xasenh xacosh

ax dxxasenhx

103. ( )∫

=

2 xa tgh log

a1

dx

xasenh1

13

104. ∫ −= cotgh x dxxsenh

12

105. ( ) ( ) ( )∫ =⋅a 2

xasen 2

dx xacosxasen

106. ( ) ( ) ( )( )

( )( )∫ −−

−++

−=⋅ba 2

xba cos ba 2

xba cos dx xbcosxasen

107. ( ) ( ) ( )( )

( )( )∫ ++

−−−

=⋅ba 2

xbasen ba 2

xbasen dxxbsenxasen

108. ( ) ( ) ( )( )

( )( )∫ ++

+−−

=⋅ba 2

xbasen ba 2

xbasen dxxbcosxacos

109. 2

xsen 2

∫ = dx senx cosx

110. ∫ =3

xsen 3

xx cosx dsen2

111. ∫ −=3

xcos 3

x sen x dxcos 2

112. 1n

xsen 1n

+=

+

∫ dxcosxxsenn

113. 1n xcos

1n

+−=

+

∫ dxsenxxcosn

114. ∫ = x tg log

dxcosxsenx

1

115. ∫ −= xcotg x tg dxx x cossen

122

116. ∫ +=− xcotg xtg

dxxcosxsenxcosxsen

22

22

117. ∫ = xcos

1 dx xcos

senx2

118. ∫ −=sen x

1 dxxsenxcos

2

14

119. ( ) ( ) ( )[ ]∫ −= xlog cos xlogsen2x dxlog xsen

120. ∫ +−=+

xcos1 2dxxcos1

cosxsenx2

121. x2cos

1 2=∫ dxxcos

senx3

122. ∫ +−

−−−= 1sen x1sen x log

21sen x

3xsen

3

dxcosx

xsen4

123. ( )∫

+++−=

+ 31 x tg2 arctg

31cosx senx 1 log

21

dx

1cosxsenxxsen2

124. ( ) ( ) xcossen xarcsen

21 2x sen xcossen x log

21

dx xtg

1dx x cotg

−+++=

===∫ ∫∫dx senxcosx

125. ∫ −= x cos 2

dxxcos

senx

126. ( )∫ −

+=

− sen x x cosx xcossen x x

dx

xsenxcosxx

2

2

127. ∫ −−=+−

2x xcos xcossen x

21 dx

sen x1xsensen x 3

128. ∫ +=−

sen xx

dxxcos1

xsen2

129. ∫ +=+

cosxsenx dxcosxsenx

2xcos

130. ∫ −=+ x cos log 2 xtg dx

xcossen2x1

2

131. ∫

++=

++

2xtg1 log

2xtg 2 dx

cosx1senx1

132. ∫ =++

2x tge xdxe

cosx1senx1 x

15

133. ∫ −=+− x

2x tg2 dx

cosx1cosx1

134. ( )∫ +=+− x cos x log dx

cosxxsenx1

135. ∫

+

=−+

xcossen x1x

dxxsen1xcosx

136. ( )∫ =+

sen x arctg dx xsen1

cosx2

137. ( ) ( )∫

±=

±

8π

2ax tg log

2 a1

dx

xacosxasen1

138. ( )( ) ( ) ( ) ( )∫ ±+±=

± xacos xasen log

a 21

2x

dx

xacosxasenxacos

139. ( )( ) ( ) ( ) ( ) x a cos x asen log

a 21

2x

±=±∫ mdx

xacosxasenxasen

140.

)

)

∫

−+

⋅

−+

+

−⟩

⋅

+

−⟩

=+

abab

2xtg

abab

2xtg

log ab

1 a b se 2

2x tg

b-aba arctg

ba1 b a se 1

22

22

dxxcosba

1

141. ∫

+−−

++−

+=

+ 22

22

22bab

2x tga

bab2x tga

log ba

1

dxsenxbcosxa

1

142. ( ) ( )∫ −= xa cos loga1 dxaxtg

143. x xtg −=∫ dxxtg 2

144. x xtgxtg31 3 +−=∫ dxxtg 4

16

145. ( )

( )∫ ∫∫ −−−

== dxx tg1nxtgdx

xcotg1 2n

1-n

n x dxtg n

146. ( ) ( )( )1na

xatg 1

1n

2 +=

+

+

∫ dxxsen

xtg n

147. ( ) 1x tg2 log41 2 +=

+∫ dxxsen1

tgx2

148. xtg4 log 4 xtg +−=++∫ dx

4tg xtg xxtg 3

149. ( ) ( ) xasen log a1 =∫ dxxacotg

150. ∫ −−= x xcotg dxxcotg 2

151. ( )( )∫ ∫ ∫−−

−== dxx cotg1n

xcotgdxxtg

1 2-n1-n

ndxxcotg n

152.

)

)∫

−+

⟨

−−

⟩

=

22

22

axaxarccosh x 0

axarccosh per 2

axaxarccosh x 0

axarccosh per 1

dxaxarccosh

153. ∫ +−

=

22 a x

axarcsenh x dx

axarcsenh

154. ( ) ( ) x a cosh log a1 =∫ dxxatgh

155. xa log 2a

axarctgh x 22 −+

=

∫ dx

axarctgh

156. a x log2a

ax arccotgh x 22 −+

=

∫ dx

axarccotgh

157. ( ) ( ) x a senh log a1 =∫ dxxacotgh

158. ( ) ( ) x a sen a1 =∫ dxxacos

17

159. ( ) ( ) ( ) x a senax x a cos

a1 2 +=∫ dxxacosx

160. senx 2cosx x 2senxx 2 −+=∫ dxxcosx 2

161. ( ) x cos 2xsen x 2 +=∫ dxxcos

162. ( )2x xcossen x

21 x 2 sen

41

2x +=+=∫ dxxcos 2

163. xsen31 sen x sen x

32 sen x x cos

31 32 −=+=∫ dxxcos 3

164. ( )∫

+=

4π

2 xa tg log

a1

dx

xacos1

165. ( )

=

+∫ 2 xa tg

a1

dx

xacos11

166. ( )

−=

−∫ 2 xa cotg

a1

dx

xacos11

167. xtg =∫ dxxcos

12

168. x tg31 xtg 3+=∫ dx

xcos1

4

169. ( ) ( ) ∫∫ −− −−

+−

= dx xcos

1 1n2n

xcos 1nsen x 2n1ndx

xcos1

n

170. ∫∫ −− −+= dx x cos

n1n sen x x cos

n1 2n1ndxxcosn

171. ( )∫

+

=

+

2 xa cos log

a2

2 xa tg

ax

2dxxacos1

x

172. ( )∫

+

−=

−

2 xasen log

a2

2 xa cotg

ax

2dxxacos1

x

173. ( )( )

−=

+∫ 2 xa tg

a1 x

dx

xacos1xacos

18

174. ( )( )

−−=

−∫ 2 xa cotg

a1 x

dx

xacos1xacos

175. ∫ += x cos log xx tg dxxcos

x2

176. ( )∫ = 2 x tg21 dx

xcosx

22

177. dxsen x xmsen xx 1mm ∫∫ −−=dxxcosx m

178. ( ) ( ) ( )[ ]∫ += xlog cos xlogsen 2x dxlogxcos

179. ( ) ( ) xa cos a1 −=∫ dxxasen

180. ( ) ( ) ( )∫ −= xa cos ax xasen

a1 2dxxasenx

181. cosx 2senx x 2cosx x 2 ++−=∫ dxsenxx 2

182. cosxsenx 21

2xsen2x

41

2x

−=−=∫ x dxsen2

183. xcos31 xcos 3+−=∫ dxxsen3

184. ( )∫

=

2 xa tg log

a1 dx

axsen1

185. ( )∫

−=

+ 4π

2 xa tg

a1

dx

xasen11

186. ∫ +−=

+2xtg1

2 dxsenx11

187. ( )∫

+=

− 4π

2 xa tg

a1

dx

xasen11

188. ∫ −=

−2xtg1

2 dxsenx11

19

189. ( )∫

−+

−=

+ 4π

2 xa cos log

a2

4π

2 xa tg

ax

2dxxasen1

x

190. ( )∫

−+

−=

− 2 xa

4πsen log

a2

2 xa

4π cotg

ax

2dxxasen1

x

191. ( )( )∫

−+=

+ 2 xa

4π tg

a1x

dx

xasen1xasen

192. ( )( )∫

++−=

− 2 xa

4π tg

a1x

dx

xasen1xasen

193. ( )∫ ∫ −− −−

+−

−= dx xsen

11n2n

xsen 1n xcos 2n1ndx

xsen1

n

194. ∫ −= xcotg dxxsen

12

195. ∫∫ −− −+−= dxx sen

n1ncosxx sen

n1 2n1ndxxsenn

196. cosxxcos 32xcos

51 35 −+=∫ dxxsen5

197. dxcosx xncosxx 1nn ∫∫ −+−=dxsenxx n

198. ( ) ( ) ( ) ( )∫ ∫∫ −−==+ cosx dxcos1dxsenx xsen n 2n 2 dxxsen 1n2 si sviluppa il binomio e si

arriva ad integrali del tipo ( )( )

∫ +=

+

1n 2 x cos xcos dx cos

1n 2n 2

199. ( ) ( ) ( ) ( )∫ ∫∫ −==+ x cos dx cosxcos1dxx cossen x xsen nm2nm2 dxxcosxsen n1m2 si

riduce ad integrali del tipo ( ) ( )( )

1n2mxcos cosx dx cos

1nm 2nm 2

++=

+++∫

200. ( ) ( ) ( ) ( )∫ ∫ −= dx xsen1 x sen n2m 2 dxxcosxsen n2m2 si riduce ad integrali del

tipo∫ dxx sen p 2

20

201. ( )( )

∫ ∫ ∫= +

=

= dt t sen

21dx x2sen

21 n

1n

t2x ponendon

dxxcosxsen nn

202. 22 xaaxarcsen x −+

=

∫ dx

axarcsen

203. x2arcsen x x1 2xarcsenx 22 −−+=∫ dxxarcsen2

204. ( ) 2223 x1 6 xarcsen x 6 xarcsen x1 3xarcsenx −+−−+=∫ dxxarcsen3

205. 2xx 21xarcsen

21x −+

−=∫ dxxarcsen

206. 2222

xa 4x

axarcsen

4a

2x

−+

−=

∫ dx

axx arcsen

207. ( ) ( ) 422

x1 21xarcsen

2x

−+=∫ dx xx arcsen 2

208. ∫∫ −−=

−

dx x1

xarcsenx n x arcsenx 2

1nndxxarcsenn

209. ∫ −−

=

22 xa

ax arccosx dx

axarccos

210. ∫ −−

−=

22

22

xa 4x

ax arccos

4a

2x dx

axarccosx

211. ( )22 xa log 2a

ax arccotgx ++

=

∫ dx

axarccotg

212. ( )2 xa

ax arccotg ax

21 22 +

+=

∫ dx

axarccotgx

213. ( )22 xa log2a

ax arctgx +−

=

∫ dx

axarctg

214. ( )2 xa

ax arctg ax

21 22 −

+=

∫ dx

axarctgx

215. ( )∫ ++−+

= 222

x1 log21 xarctg x xarctg

21xdxx arctgx 2

21

216. ( )∫ −+= xx arctg 1x dxxarctg

217. ∫ +−

+−

=

+− 2x1 log

1x1x arctgx dx

1x1xarctg

218. ( )∫ =+

x arctg log dx arctg xx11

2

219. ( )( )

( )( )1n a

xacotg 1n

+−=

+

∫ dxxasenxacotg

2

n

220. ∫ =+

2 xarctg 21 dx

x1x

4

221.

++

−++

=−∫ 3

1 x2 arctg 3

1x1

1xx log 31

2

dxx1

13

222. ∫ = xlog dxx1

223. x xlog x −=∫ dxxlog

224. ( )2 xlog 2xlogx 2 +−=∫ dxxlog 2

225. ( )∫ ∫ −−= dxx lognxlogx 1nndxxlog n

226. ( ) ( ) ( )∫ +−+

+=+

b xba xba log

b xba dxxbalog

227. ( ) ( ) x2 x logx 2 −=∫ dxxlog 2

228.

−=∫ 2

1 xlog 2

x 2

dxxlogx

229. ∫

−=

3x xlogx

31

33dxxlogx 2

230. ∫ −=16x xlog

4x

44

dxxlogx 3

22

231. ∫

+−

+=

+

1n1 xlog

1nx

1n

dxxlogx n

232. ∫ = xlog 21 2dx

xxlog

233. ∫ −−=x1

x xlog dx

xxlog

2

234. ∫ = xlog 41 4dx

xxlog 3

235. ( )

∫ +=

+

1nxlog

1n

dxx

xlog n

236. ∫ = x log log

dxxlogx

1

237. ∫ = xlog 2

dxxlogx

1

238. ( ) ( )[ ]1 xloglog x log −=∫ dx

xxloglog

239. ( )∫ =−

xlogarcsen

dxxlog1x

12

240. ( ) ( )2xarcsen x

21x1x1 logx −+−++=−++∫ dxx1x1log

241. ( ) ( ) ( )[ ]∫ −= xlog cos xlogsen 2x dxxlogsen

242. ( ) ( ) ( )[ ] xlog cos xlogsen2x +=∫ dxxlogcos

243. ( ) ( ) xlogsen =∫ dxxlogxcos

244. ( ) ( )axarctg 2a2xaxlogx 22 +−+=+∫ dxaxlog 22

245. ( ) ( )

−+

+−−=−∫ axax log a2xaxlogx 22dxaxlog 22

23

246. ( ) ( )∫ +−++=++ 22 x1x1x logx dxx1xlog 2

247.

−=∫ 5

3 xlog x 53 3 5dxxlogx3 2

248. ( ) ( ) ( )∫ −+++=+ xxxx ee1loge1loge dxe1loge xx

249. ( )∫ −= elogxlogx aadxxloga

250. ( )∫ ∫ −

+

−−

+−

+−

=

+

dx 1n2x1

122n32n

1n2x1 22n

x dxn2x1

1

251. ∫

+

−

+

−−

−+

−=

+

−=

=++

c4qp

p2xsetttgh4qp

2

:ottiene si e radici le trovanosi

∆p2xarctg

∆2

2px

1

22

dxqpx2x

1

252.

+

−

+=

+

−

+=

=++∫

c4qpp2xsettcosh

cp4qp2xsettsenh

1

2

2

2dx

qpxx

Caso ∆ = 0

Caso ∆ < 0

Caso ∆ > 0

Caso ∆ < 0

Caso ∆ > 0

24

253. ∫ +

+==

++−c dx

qpxx

12 2p4q

p-2xarcsen

254.

( )∫

−+

−−

+++=

−−+

+−−+

=

=

=++

+

∆p2x arctg

∆ap2bqpxx log

2a

xx logxxbaxxx log

xxbax

xe xse

2

212

21

21

1

21

hasiradicilesono

dxqpxx

bax2

255. ( ) ( ) ∫∫ ++−++=

++dx

cbxax1

2abcbxaxlog

2a1 2

2dxcbxax

x2

256. ( )

riconduce si che

∆p2x1

∆p2xd

∆2 n2

12n

∫ ∫

−+

+

−+

•

−

=++

−

dxqpxx

1n2

ad integrali del

tipo: ∫

+

dt

t

an2

1

1

257. ( ) ( ) ( )∫ ∫ ∫ ++

−+

++

+=

++

+ dx qpxx

12apbdx

qpxxp2x

2a n2n2

dxqpxx

baxn2

258. ∫

−−

−=

=−

=⟩∆⟨

++++=⟩

=++

1

2

21

2

xxxx arctg

a2 radici le sono

xe xse ∆

b xa 2arcsen

a1 0 , 0

acx

abx

2abx log

a1 0

acaso

acaso

dxcbxax

12

259. ( ) ( ) ( )1-2n ! 1nx1

5 ! 2x

! 13xx

12n1n

0

53

−−+⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+

⋅+

⋅−=

−−−∫

x

x dxe2

Caso ∆ > 0

Caso ∆ < 0

Caso ∆ > 0

25

260. ∫+∞

− =

0

2πdxe

2x

261. ( ) ( ) ( )∫ −+⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+

⋅+

⋅−=

−−

x

dxx

senx

0

12n1n

53

1-2n! 12nx1-

5 ! 5x

3 ! 3xx

262. ( )∫ −+⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅−+−=

1

0

n1n

32 n11-

31

211dxx x

263.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+

××××

+

××

+

+=

−∫ k642531k

4231k

211

2π 6

24

22

22

0

π

dxxsenk1

122

dove -1 < k < 1

264.

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅+

××××

−

××

−

−=−∫ k

642531

51k

4231

31k

211

2π 6

24

22

2

dxxsenk12π

0

22

265. ( )( )∫ ∫ +×⋅⋅⋅⋅⋅⋅×××××⋅⋅⋅⋅⋅⋅××××

== ++2

0

2π

0

12n

12n97532n8642 dx x cos

π

dxxsen 12n

266. ( )( )∫ ∫ ××⋅⋅⋅⋅⋅⋅×××

−×⋅⋅⋅⋅⋅⋅×××==

2

0

2π

0

2n

2π

2n64212n531dx x cos

π

dxxsen2n