Anexo D Tabla de Integrales

Anexo D
Tabla de Integrales
(PUEDE SUMARSE UNA CONSTANTE ARBITRARIA A CADA INTEGRAL)
1.
2.
3.
4.
5.
6.
7.
8.
9.
Z
Z
Z
Z
Z
Z
Z
Z
Z
xn dx =
1
xn+1
n+1
(n �= −1)
1
dx = log | x |
x
ex dx = ex
ax dx =
ax
log a
sen x dx = − cos x
cos x dx = sen x
tan x dx = − log |cos x|
cot x dx = log |sen x|
Ø
µ
∂Ø
Ø
Ø
1
1
sec x dx = log |sec x + tan x| = log ØØtan
x + π ØØ
2
4
227
228
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Tabla de Integrales
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Ø
Ø
Ø
1 ØØ
Ø
csc x dx = log |csc x − cot x| = log Øtan xØ
2
arcsen
x
x √ 2
dx = x arcsen
+ a − x2
a
a
arccos
x
x √
dx = x arccos − a2 − x2
a
a
arctan
(a > 0)
(a > 0)
°
¢
x
x a
dx = x arctan − log a2 + x2
a
a 2
sen2 mx dx =
1
(mx − sen mx cos mx)
2m
cos2 mx dx =
1
(mx + sen mx cos mx)
2m
(a > 0)
sec2 x dx = tan x
csc2 x dx = −cot x
Z
senn−1 x cos x n − 1
sen x dx = −
+
senn−2 x dx
n
n
Z
cosn−1 x sen x n − 1
n
cos x dx =
+
cosn−2 x dx
n
n
Z
tann−1 x
n
tan x dx =
− tann−2 x dx (n �= 1)
n−1
Z
cotn−1 x
n
cot x dx =
− cotn−2 x dx (n �= 1)
n−1
Z
tan x secn−2 x n − 2
n
sec x dx =
+
secn−2 x dx (n �= 1)
n−1
n−1
Z
cot x csc n−1 x n − 2
n
csc x dx =
+
cscn−2 x dx (n �= 1)
n−2
n−1
n
senh x dx = cosh x
cosh x dx = senh x
229
26.
27.
28.
29.
30.
31.
32.
Z
Z
tanh x dx = log |cosh x|
coth x dx = log |sen hx|
Z
sech x dx = arctan (senh x)
Z
1
1
senh2 x dx = senh 2x − x
4
2
Z
Z
Z
Z
Ø
x ØØ
1
cosh x + 1
Ø
csch x dx = log Øtanh Ø = − log
2
2
cosh x − 1
1
1
cosh2 x dx = senh 2x + x
4
2
sech2 x dx = tanh x
x
x √
dx = xsenh−1 − x2 − a2 (a > 0)
a
a
√
£
° ¢
§
Ω
Z
xcosh−1 xa − √x2 − a2 £cosh−1 ° xa ¢ > 0, a > 0§
−1 x
34.
cosh
dx =
xcosh−1 xa + x2 − a2 cosh−1 xa < 0, a > 0
a
Z
Ø
Ø
x
x a
35.
tanh−1 dx = xtanh−1 + log Øa2 − x2 Ø
a
a 2
Z
≥
¥
√
1
x
√
36.
dx = log x + a2 + x2 = sen h−1
(a > 0)
a
a2 + x2
Z
1
1
x
37.
dx = arctan
(a > 0)
2
2
a +x
2
a
Z √
x√ 2
a2
x
38.
a2 − x2 dx =
a − x2 + arcsen
(a > 0)
2
2
a
Z
° 2
¢3
¢√
x° 2
3a4
x
39.
a − x2 2 dx =
5a − 2x2
a 2 − x2 +
arcsen
(a > 0)
8
8
a
Z
1
x
√
40.
dx = arcsen
(a > 0)
a
a2 − x2
Ø
Ø
Z
Øa + xØ
1
1
Ø
41.
dx =
log ØØ
a2 − x2
2a
a − xØ
33.
senh−1
230
42.
Tabla de Integrales
Z
1
(a2 − x2 )
Z √
3
2
dx =
a2
√
x
a2 − x2
Ø
Ø
√
x√ 2
a2
Ø
Ø
2
2
2
43.
± dx =
x ± a ± log Øx + x ± a Ø
2
2
Z
Ø
Ø
√
1
x
Ø
Ø
√
44.
dx = log Øx + x2 − a2 Ø = cosh−1
(a > 0)
a
x2 − a 2
Ø
Ø
Z
Ø x Ø
1
1
Ø
Ø
45.
dx = log Ø
x(a + bx)
a
a + bx Ø
Z
x2
a2
√
3
2 (3bx − 2a) (a + bx) 2
46.
x a + bx dx =
15b2
Z √
Z
√
a + bx
1
√
47.
dx = 2 a + bx + a
dx
x
x a + bx
√
Z
x
2 (bx − 2a) a + bx
√
48.
dx =
3b2
a + bx
Ø√
Ø
8
Ø a+bx−√
aØ
Z
<
√1 log Ø √
√
Ø (a > 0)
1
a
a+bx+
q a
√
49.
dx =
: √2 arctan a+bx (a > 0)
x a + bx
−a
−a
50.
51.
Z √
Z
Z
√
Ø
Ø
√
Ø a + a2 − x2 Ø
a2 − x2
Ø
dx = a2 − x2 − a log ØØ
Ø
x
x
√
¢3
1°
x a2 − x2 dx = − a2 − x2 2
3
√
¢√
x° 2
a4
x
x2 a2 − x2 dx =
2x − a2
a2 − x2 + arcsen
8
8
a
√
Ø
Ø
Z
Ø a + a 2 − x2 Ø
1
1
Ø
Ø
√
53.
dx = − log Ø
Ø
a
x
x a2 − x2
Z
√
x
√
54.
dx = − a2 − x2
a2 − x2
Z
x2
x√ 2
a2
x
√
55.
dx = −
a − x2 + arcsen
(a > 0)
2
2
2
2
a
a −x
Ø
Ø
Z √ 2
Ø a + √x2 + a2 Ø
√
x + a2
Ø
Ø
56.
dx = x2 + a2 − a log Ø
Ø
Ø
Ø
x
x
52.
(a > 0)
231
57.
Z √
Z
≥x¥
√
√
x2 − a 2
a
dx = x2 − a2 − a arccos
= x2 − a2 − arcsec
x
|x|
a
(a > 0)
√
¢3
1° 2
x x2 ± a2 dx =
x ± a2 2
3
Ø
Ø
Z
Ø
Ø
1
1
x
Ø
√
√
59.
dx = log ØØ
a
x x2 + a2
a + x2 + a2 Ø
Z
1
1
a
√
60.
dx = arccos
(a > 0)
a
|x|
x x2 − a 2
√
Z
1
x2 ± a2
√
61.
dx = ±
a2 x
x2 x2 ± a2
Z
√
x
√
62.
dx = x2 ± a2
x2 ± a 2
Ø
Ø
(
Z
2
Ø 2ax+b−√
Ø
√ 1
√b −4ac Ø (b2 > 4ac)
log
1
Ø
2 −4ac
2 −4ac
b
2ax+b+
b
63.
dx =
ax2 + bx + c
√ 2
arctan √2ax+b
(b2 < 4ac)
4ac−b2
4ac−b2
Z
Z
Ø 2
Ø
x
1
b
1
Ø
Ø
64.
dx =
log ax + bx + c −
dx
2
2
ax + bx + c
2a
2a
ax + bx + c
(
√ √
Z
√1 log |2ax + b + 2 a ax2 + bx + c| (a > 0)
1
a
√
65.
dx =
−2ax−b
√1 arcsen √
(a < 0)
ax2 + bx + c
−a
b2 −4ac
58.
Z √
Z
2ax + b √ 2
4ac − b2
1
√
66.
+ bx + c dx =
ax + bx + c +
dx
4a
8a
ax2 + b + c
√
Z
Z
x
ax2 + bx + c
b
1
√
√
67.
dx =
−
dx
2
2
a
2a
ax + bx + c
ax + bx + c
Ø √√ 2
Ø
(
Z
Ø 2 c ax +bx+c+bx+2c Ø
−1
√
log
1
Ø
Ø (c > 0)
x
c
√
68.
dx =
bx+2c
√1 arcsen
√
(c < 0)
x ax2 + bx + c
−c
|x| b2 −4ac
ax2
∂q
1 2
2 2
69.
x
+ dx =
x − a
(a2 + x2 )3
5
15
q
Z √ 2
2
∓
(x2 ± a2 )3
x ±a
70.
dx =
x4
3a2 x3
Z
sen(a − b)x sen(a + b)x
71.
sen ax sen bx dx =
−
2(a − b)
2(a + b)
Z
3
√
x2
a2
µ
°
a2 �= b2
¢
232
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
Tabla de Integrales
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
sen ax cos bx dx =
cos(a − b)x cos(a + b)x
−
2(a − b)
2(a + b)
cos ax cos bx dx =
sen(a − b)x sen(a + b)x
−
2(a − b)
2(a + b)
°
a2 �= b2
°
¢
a2 �= b2
¢
sec x tan x dx = sec x
csc x cot x dx = −csc x
Z
cosm−1 x senn−1 +x m − 1
cos x sen x dx =
+
cosm−2 x senn x dx =
m+n
m+nZ
senn−1 x cosm+1 x
n−1
= −
+
cosm x senn−2 x dx
m+n
m+n
Z
1 n
n
n
x sen ax dx = − x cos ax +
xn−1 cos ax dx
a
a
Z
1 n
n
n
x cos ax dx = x sen ax −
xn−1 sen ax dx
a
a
Z
xn eax n
n ax
x e dx =
−
xn−1 eax dx
a
a
∑
∏
1
n
n+1 log ax
x log(ax) dx = x
−
n+1
(n + 1)2
Z
xn+1
m
m
m
n
x (log ax) dx =
(log ax) −
xn (log ax)m−1 dx
n+1
n+1
m
n
eax sen bx dx =
eax (a sen bx − b cos bx)
a2 + b2
eax cos bx dx =
eax (b sen bx + a cos bx)
a2 + b2
sech x tanh x dx = −sech x
csch x coth x dx = −csch x