Formulas de integracion

ax + b
(1)
Z
(ax + b)ndx =
1
a(n + 1)
(ax + b)n+1 + C; n 6= 1
(2)
Z
dx
ax + b
=
1
a
ln jax + bj + C
(3)
Z
dx
x(ax + b)
=
1
a
ln
____
x
ax + b
____
+ C
(4)
Z
dx
(1 + _x)2 =
1
_ _
1
1 + _x
+ C
(5)
Z
xdx
(1 + bx)3 =
1
2b _
2
(1 + bx)2
1
2b _
1
1 + bx
+ C
1.1.2. Contienen pax + b
(6)
Z
xpa + bxdx =
2(3bx 2a)(a + bx)3=2
15b2 + C
(7)
Z
xpa + bx
dx =
2(bx 2a)pa + bx
3b2 + C
1
(8)
Z
dx
xpa + bx
=
8<
:
1 pa ln
___
pa+bxpa pa+bx+pa
___
+ C; a > 0
2 pa arctan
q
a+bx
a + C; a < 0
(9)
Z pa + bx
x
dx = 2pa + bx + a
Z
dx
xpa + bx
+ C
1.1.3. Contienen x2 _ a2
(10)
Z
dx
a2 + x2 =
1
a
arctan
x
a
+ C; a > 0
(11)
Z
xdx
(x2 _ a2)3=2 =
1
px2 _ a2
+ C
1.1.4. Contienen a2 x2; x < a(12)
Z
(a2 x2)3=2dx =
x
2
p
a2 x2 +
a2
2
arc sen
x
a
+ C; a > 0
(13)
Z
dx
a2 x2 =
1
2a
ln
____
a + x
a x
____
+ C =
1
a
arctanh
x
a
(14)
Z
dx
(a2 x2)3=2 =
x
a2pa2 x2
+ C
1.1.5. Contienen px2 _ a2
Z p
x2 _ a2dx =
1
2
_
x
p
a2 _ x2 + a2 ln
__ _
x
+
p
a2 _ x2
___
_
(15) + C =
=
(
1
2xpa2 + x2 + a2
2 arcsenhx + C (+)
1
2xpa2 x2 + a22 arccoshx + C ()
(16)
Z
x
p
x2 _ a2dx =
1
3
(x2 _ a2)3=2 + C
(17)
Z
x3
p
x2 + a2dx = (
1
5
x2
2
5
a2)(a2 + x2)3=2 + C
(18)
Z px2 a2
x
dx =
p
x2 a2 a _ arc cos
a
jxj
+ C
(19)
Z
dx
px2 + a2
= a _ arcsenh
x
a
+ C
(20)
Z
dx
px2 a2
= ln
___
x +
p
x2 a2
___
+ C = arccosh
x
a
+ C; (a > 0)
(21)
Z
dx
xpx2 a2
=
1
a
arc cos
a
jxj
+ C;(a > 0)
2
(22)
Z
dx
x2px2 _ a2
= _
px2 _ a2
a2x
+ C
(23)
Z
xdx
x2 _ a2 =
p
x2 _ a2 + C
(24)
Z px2 _ a2
x4 dx = _
(a2 + x2)3=2
3a2x3 + C
(25)
Z
x2
px2 a2
dx =
x
2
p
x2 a2
a2
2
arccosh
x
a
+ C
1.1.6. Contienen pa2 _ x2
(26)
Z p
a2 x2dx =
1
2
x
p
a2 x2
a2
2
arc sen
x
a
+ C; (a > 0)
(27)
Z
x
p
a2 _ x2dx = _
1
3
(a2 _ x2)3=2 + C(28)
Z
x2
p
a2 x2dx =
x
8
(2x2 a2)
p
a2 x2 +
a2
8
arc sen
x
a
+ C; a > 0
(29)
Z pa2 _ x2
x
dx =
p
a2 _ x2 a ln
_____
a + pa2 _ x2
x
_____
+ C
(30)
Z
dx
x2pa2 x2
= pa2 x2
a2x
+ C
(31)
Z
dx
pa2 x2
= arc sen
x
a
+ C; a > 0
(32)
Z
dx
xpa2 x2
=
1
a
ln
_____
a + pa2 x2
x
_____
+ C
(33)
Z
x
pa2 _ x2
dx = _
p
a2 _ x2 + C
(34)
Zx2
pa2 _ x2
dx = _
x
2
p
a2 _ x2 _
a2
2
arc sen
x
a
+ C; a > 0
(35)
Z
dx
pa2 + x2
= ln
_
x +
p
a2 + x2
_
+ C = arcsenh
x
a
+ C; a > 0
1.1.7. Contienen ax2 + bx + c
(36)
Z
dx
ax2 + bx + c
=
8>>>><
>>>>:
1 pb24ac
ln
___
2ax+bpb24ac
2ax+b+pb24ac
___
=
= 2 pb24ac
arctanh 2ax+b pb24ac
+ C; b2 > 4ac
2 p4acb2 arctan 2ax+bp4acb2 + C; b2 < 4ac
2
2ax+b + C; b2 = 4ac
3
(37)
Z
x
ax2 + bx + c
dx =
1
2a
ln
__
ax2 + bx + c
__

b
2a
Z
dx
ax2 + bx + c
+ C
(38)
Z
x _ dx
(ax2 + bx + c)n =
bx + 2c
(b2 4ac)(n 1)(ax2 + bx + c)n1+
+
b(2n 3)
(b2 4ac)(n 1)
Z
dx
(ax2 + bx + c)n1 ; n 6= 0; 1; b2 < 4ac
(39)
Z
dx
(ax2 + bx + c)n =
2ax + b
(b2 4ac)(n 1)(ax2 + bx + c)n1+
+
2a(2n3)
(b2 4ac)(n 1)
Z
dx
(ax2 + bx + c)n1 ; n 6= 0; 1; b2 < 4ac
1.1.8. Contienen pax2 + bx + c
(40)
Z p
ax2 + bx + cdx =
2ax + b
4a
p
ax2 + bx + c+
4ac b2
8a
Z
dx
pax2 + bx + c
(41)
Z
a0 + a1x + : : : + anxn
pax2 + bx + c
dx Ver x3.5, p_ag. 11: m_etodo alem_an
(42)
Z
dx
pax2 + bx + c
=
1
pa
ln
___
2ax + b + pa
p
ax2 + bx + c
___
+ C =
8><
>:
1 paarcsenh 2ax+b p4acb2 + C; _ < 0; a > 0;
1 pa ln j2ax + bj + C; _ = 0; a > 0;
1
pa arc sen 2ax+b pb24ac
+ C; _ > 0; a < 0;
; _ = b2 4ac
(43)
Z
x
pax2 + bx + c
dx =
pax2 + bx + c
a
b
2a
Z
dx
pax2 + bx + c
(44)
Z
dx
xpax2 + bx + c
=
8<
:
1 pc ln
___
2pcpax2+bx+c+bx+2c
x
___
+ C; c > 0
1 pc arc sen bx+2c
jxjpb24ac
+ C; c < 0
1.2. Funciones...