Manuale di Matematica: Tabella Integrali Fondamentali
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Transcript of Manuale di Matematica: Tabella Integrali Fondamentali
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: [email protected]
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