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INSTITUTO POLITÉCNICO NACIONALEscuela Superior de Cómputo
“ Lista de Integrales ”
Alumno:Yosef Israel Gutiérrez Moreno
Grupo:1CV9
Profesor:Alfredo Guzmán Rangel
Materia:“Calculo”
Semestre:14/2
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2LISTA DE INTEGRALES1)
=6
2.
(√ )
√ =223
22√3 3)
3√
√ 4
= 3√ √ 4 = 3− /4=3− /14 /
=3 /1/214 /5/2
=6/ √
4. ∫√ = =· ∫ 2 = =√ 2∫
=2
=√ 2+
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3=√ 2 6. ∫√
√ = = − =34 =43 =43
8.∫ u=5x; du=5 dx
= ∫ =
9.
5 =5
= 5 →5=
→ 5=15 =15 = 55 11. ∫ =∫∗ ∗=∫ ∗ == =ln
=1
∗ = 12.3 =3 =3
3=
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4= 3
=13
=13 =13 cot =cot33 14.
3 7 =3 7 =3 →3=
→ 13 =
13
=13ln|| =13ln|3 7| 15.-
∫ − u=1-x= ∫ − =
= | |
17. ∫tan2 =2 =2 =12 tan∫u12 ∫tan2
12 tan
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5∫tan2 ∫ sincos2
=cos = sin=1sin ∫sin 1sin∫ 12
2=cos2
2 =cos22
. = == ||= | |
=5 7 → =55= = → = 19. 3= tan3 =13 tan =13cos=13 1ℎ ℎ= log3= 13logcos= 13ln|cos3| . = 3 & =3 ; ℎ=c& = sin
ɵ∗ɵ∗ ɵ u= ɵ du=
ɵ∗ ɵ
=2 = ɵ
23.
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6 454 44
=4 =4 =4 =4 14 4 14ln| | 4ln| |
14ln| 4 | 4ln4 24)cos = =3 =3 u=sen x =cos
25.
∫ =
=∫
=∫ 1 = =∫ = = + = = 27. ∫ √ + √ 2 3=
2 3=2 3
−
=23 =4 4=
− 4=14
− =1412
=24
=122 3
=12 2 3
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728.-∫ + =∫ 1− =∫ − = ∫ − = = = 1 = √
=3 =3 3=
29.∫ U=sen(x), du=cos(x) dx
=∫ = = - 30.
= = → − = − = −2 =12 32.
∫ =∫ = +=∫ =
=cot =1 =
33. 1 = 1 =
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8= 1− = − =12=2=2 1=2 1
35.
√ 2 1 =2 1 =2 →2=
→ −⁄2=12 −⁄ =121
2=⁄
=√ 2 1 36-∫ + = =
=
= = = | | 38.
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9∫ tan 1(cos)
=tan =sec=cos √ 1coscos √ 1 = 1 =1 =1
√ 1
√ +12 1=
tan1+12 1=2 13
.∫ + =∫ =∫− =− = +=2 3 → =3 26= 2 √
u=arcsenx
du= √ −
=2 = 43)
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10cot1 = = =2
=
2
= cot =11 44.
√1
=arccos , = √1
1 =3 = 2 45)
cot1 = = 2 = 2 = cot
=11 46. ∫ += =∫ + =∫ +2 = 1 =∫ =2 =∫ = =∫ + = | 1| 48.
∫
+
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11cos 2sin 3 u=2sinx 3 =2cos
2=cos 2=2=12 =12ln =12ln 2sin 3
49-∫ = ∫ = ∫ = | | = |
50.∫2 1 =2∫ 1 U= +1 , du= 2x dx
=
∫du
= + c
= ( 1 + c51.
= = 1 = = 1 =
= =
= = =
=
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12= =3
=3
52.-∫ + =tan− =1 1 = = ln= lntan− 53. ∫ +=∫ =ln =ln3 1 =3 tan1
=3
3=1
54. = =
= = =4 =4 56.
cos22 3 2=2 3 2
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13=6 2 →6= 2 →6=16
=16ln|| =16ln|2 3 2| 57.-
∫ = ==
= = 59)
∫ ∫
∫+
1 1
∫2=2 . = = =2=12 → =12 2 =
61.
cos = = = . =
U=
Du= =
=
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14= = 64)
∫ ∫ ∫ +1 1 ∫ = =1
4
65.
3
=3 =3ln3
1ln3ln3 1ln3ln3 =1ln3ln3
∫ 1ln3 =1ln3 11ln3 1 3 +1ln3 1 ln3
3ln3 166)
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15− =3=13 =13 =1
3−
= 3 = 3 67. ∫ = ∫ ∫
=5 =∫ ∫ =5 =∫ ∫= = ∫ =5 = ∫ =5 = ∫ = = = =
69. ∫ − = 2
= 2 = 2
= 2 =lnln 2=ln lnln ln2=ln lnln ln2 =ln ln2
70.-∫ + =∫ + =∫ =∫ = ∫ = | |
=
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16= =
= 71.∫ + U= 2x, du=2dx
= ∫ + S= 2, ds= du
=
∫
=
= log( 2 = log( 2 72.
1 2
=1 ; =√ 2 =√ 2 →√ 2= →1√ 2 1 =1√ 21−
=1√ 2
−√ 2
73.- ∫√ − =∫√ ·√ ·√ =√ 3 =√ 3 →√ 3= √ 3√11
= √ 3 · √1=
13√1=1√ 3sin− C
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1774.√16 9=13√ =13 =13 394
√=
=16, =4: =9, =3 75.√9 =9 ; =3 = ; = =1 =1
3 377.9 4 =3 ; =2
=3 →3=
13 =13 1312arctan32 =16 arctan32 78.-
= =
= =
[ ]
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18
| | | |
| | | | | |
80.
∫√ =asec =tansec = sec =atan =arcsec
12∫sec sectansec tan
s=tanu secu ds=sectansec 1∫1s =
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2087)
√3 5=1√ 3
1 153
=1√ 5
1√1
= 1√ 5
= 53√ 5
= 53 = 53 90. ∫ −√ − cos−√1 =cos−√1 √1 =cos−√1 1=cos−√1 1− =cos− , =1
=√1, = 2
=√1, 2= − 2 = 12 − =21212=12cos− 221 =12cos− 1
91.-∫ −+ = ∫ − = ∫ + =∫ = ∫ = 1 =∫ = ∫ = = =
=
= =
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2295 .∫ +=arctanc Se utiliza la formula += = , = = ; =, =
96.cos √ = =cos = −cos
= − =13 =3=3 =3√
98.
2√ 1 :
2 =2 →2√ 1
= 1 = 2 → − = 2 = 2 1 99.-
∫ = = ∫ =
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23
− −
= 101.
∫2sin 3cos =? 122.
∫ 2√ ∫ = = 2 ∫1√
√
= . = ==
103.∫ − +=∫√ −√ + =√ ∫ + =
√ ∫ + =√ ∫ + =√ ∫ + =√ tan− =
√ =√ 3 √ , =√ 3 ; = , =
6 9 9 5
34
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24= 3 =
∫ −=
ln−+
=12 2ln 3 232 =
106)
2 2 1= = =
111. ∫ ++ − 7 16 1=7 12 1 3 1 7 12 13 1= 2 13 1 7 1=3 12 1 7 1=3 2
7 1=3 2 3 2 =7 =1 =1 =2 7 12 13 1=12 123 1
7 12 1 3 1=12 123 1 =2 12 3 1
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25=2 1, =3 1 =2 , =3
2= ,3=
2 23=2 23=12 23 =12ln| |23ln| |=12ln 2 123ln 3 1 =23ln 3 112ln 2 1 112.-∫ −− + =∫ −− + = − +− + = ∫ − + =2∫ − + = √ √ 39√ √ =
√ 39 √ 39√ √ = √ √ 39√ √ ) = √ 39 √ 39√ √ = ln √ 39 √ 39√ √ = ln √ √ 10 1 = ln√ √ 10 1
117.√ 2 =ln 2 .√ = √ √ =
√
= || = = = =4 13 → =4 4 , 1 ,4
= 1 ,4 1=
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26124.∫ −√ + − = ∫√ = 2√ = √ = √ 1166 3 129)
= cos= cos = = = = = = cos
132. ∫ln 1 ln1 =ln1 , = =1 , = ln1 = ln1 1 = 1 = 1
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27= ln1 1 = ln1 1= ln1
= ln1 = ln1 ln1= 1ln1 133.-∫ = + ∫ + =∫ + = + ln++ C = =
=
135.
= =√1
= =2 →2 2√1 =2 2√1 =2 =12 = √1 = 1 →2 [2 1 1 2] =2 [2 1 12 1 ] ó = 1 =
=
1 :
= =
= 1
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28= =1 22 =212 2 =214 2=12 12 2
=12 :=→ = √1= ℎ :
2 [2 11212 1 ]=2 2 1 14 14 1 =2 4 1 14 142 1 1
136.-
∫log 1 =∫ =∫
= ln1 = =+ = = ln1∫ + = ln1 2 1
= ln1 2 11
1
= ln1 21 1 2 1 = ln1 2tan− 2 1 = ln1 2tan− 2 137. ∫arctan√ =arctan√ ∫√ + = √ =
=12√ 1 =
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29√ = =, =2 arctan√ 12 2 1 = arctan√ 1
= arctan√ 111 =
arctan√ 1 = arctan√ arctan= arctan√ √ arctan√ 138.√ √
= √ =√
=1 1√ =−
=1√ 1 =2√ =2√ √ 2√ √ 1 =2√ √ 2√ √ 1 =2√ √ 2(√ 1 ) 140.
= = = =
→ = 1 22 =12⌈ 2 =12[ 12 2 ]=12[ 22 ] =12
→2 12
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30=2 12 12 2 122 =2 2 4 2 =42 4 2 =4 24 2 143.∫x tan− 1
=√1 =√ − ∫u tan−u =tan−u = =1 1 =2
12tan−u12∫1 12∫ 1 1
12tan−uu212tan−u =12tan− 1 1 . = = =√
= √ = || = → =1√1 , =− → =
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32 csc = cos/ ℎ= ℎ=cos 1ℎ ℎ =1ℎcsc=√ 1
151. ∫√ + = ∫ =∫ =| |
√ = =√ + = = =√ + =
153. ∫ + + + 1 3 5= 1 3 5
= 3 5 1 5 1 3
= 8 156 54 3 = 8 156 54 3 = 8 6 4 =0 8 6 4 =1
15 5 3 =0
=18, =34, =58
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3318 134 358 5 =18 134 358 5=
18 1
34 3
58 5==18ln 134ln 358ln 5=18 ln16ln 35ln 5=18ln 35 1
156.
1 2
11 2= 1 1 2 1= 1 2 2 1 =1 = 1 =2 =1 3 2 22 =1 2 2 =1 2 2 1=1
2 = 2 = 1 1 1 2= ln| 1| 1 1ln| 2| =1 1ln 12
157.-∫ −− + =∫ −−
=22
23
2 = 31 2 21 2 21 =21 31 2 21 =2ln 3∫ 2∫ … 3∫ = 3= …=2ln2ln = − ln
−
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34
161.
2 3 312 5
2 3 312 5= 1 2 5 2 3 3=2 5 1 2 5 =23 3 → =2 2 = 3 5 = 3 = 1
=3 = 213 22 5
ln| 1| 2 22 512
=ln2 5
1 12
164.
∫ 3 74 4
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35
=
42 1
12
−2
196.-∫ =∫ = ∫ 1 =∫ ∗ = ∫ ∗∫ =∫ ∗∫ = ∫ = ∫ =+ = = = = =
= = =
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36
198.
= = 1 = =
5 7
199.- ∫ =∫ = ∫ 1 = = = 1 = 1
= 13
= 3 =csc3 = 13 200.∫ =∫ +=∫ ∫ =∫ ∫ =
=414 2 = =1 cos22 201.
= = [1212cos2 ]=
1412cos2
14 2
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37=4 cos22 8 cos4 8 =14
12
cos218
18
cos4 =2 =4 =2 =4
2= 4= 14 12 cos218 18 cos4 14
1212 cos
18
1814 cos
=14 14 218 132 4 =38 14 2132 4 203.
= 1 22 1 22 116218 2 116 1162 =116 2
2 =1 44 = 1 2 4 44 14 4 =14 4 4 =1 cos82
12cos8 12 =12 cos8 2
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3818 cos =18 = 1418 82 =132 816
12 =2 12 4 =12 4 =18 4 14 =4 18 2 =
18 2 2 =1 42 12 4 =18 4 12 =2 116 =16
→31281128 411024 8 →11283 4sin88
206.
∫cot− cot x ( cos² x /sen² x )cot x ( (1 – sen² x)/sen² x )((cot x/sen² x) – cot x)
∫ cot³ (x) dx = ∫ ((cot x/sen² x) – cot x) dx =∫ (cot x/sen² x) dx – ∫ cot x dx =
(-1/2) cot² x – In(sen x) + C
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40=6 = =∫ · ∫=∫ ∫ = ∫
=− ∫ =2 =2 = =− ∫ · =− ∫ =− =− 220.-∫ + =∫ + = 1 = =− = 1 1 = 1 1 =1 1