Calculo

Integral indefinida.
Se representa mediante:

∫ f  x  dx = f  x c

Formulas: Reglas básicas de integración:

∫ dx = x c
∫ k f  x  dx =k ∫ f  x  dx
x n1
x n dx =
 c n≠−1
∫
n 1

∫ k dx = k x c
∫ [ f  x ± g  x ] dx =∫ f  x  dx ±∫ g  x  dx

∫ cos x dx = sen x c
∫ sen x dx =−cos x  c

∫ sec2 x dx =tan x c
∫ csc 2 x dx =−cot x  c

∫ sec x tan x dx =sec x c
∫ csc x cot x dx =−csc x c

Ejemplos:
x 11

3

a

∫ 3 x dx =3 ∫ x 1 dx =3 1 1 c = 2 x 2c

b

∫ x 3 dx =∫ x−3 dx =−31  c = −2  c =− 2 x 2  c

c

d

x −31

1

1

1
2

1

3

1

x2
x2
2
 c =  c =  x 3 c
 x dx =∫ x dx =
∫
1
3
3
1
2
2

∫ 2 sen x dx =2 ∫ sen x dx = 2 −cos x c =−2 cos x  c
1

1

e

x−2

1

−
2
x2∫  x dx = 2∫ x 2 dx =2 1 c = 4 x 2  c =4  x  c
2

f

∫  x  2 dx =∫ x dx ∫ 2 dx = 1 x 2 2 x c
2

g

∫  3 x 4−5 x 2 x  dx = 3 x 5− 5 x 3 1 x 2c
5
3
2
1

h

1

3

1

1

−
−
x 1
x2 x2
2
dx =∫ [ x 2  x 1 ] dx =∫  x 2  x 2  dx =  c =  x 32  x c
∫ x
3
1
3
2
2

i

∫  t 21 2 dt =∫  t 4 2 t2 1 dt = 1 t 5 2 t 3t  c
5
3j

x 3 3
12
x−1
1
3
−2
3
−2
∫ x 2 dx =∫ x  x 3  dx =∫  x  3 x  dx = 2 x 3 −1  c = 2 x 2 − x  c

k

l

1
3

4
3

1
3

7

4

x3
x3
33
3
∫  x  x −4  dx =∫ x  x −4  dx =∫  x − 4 x  dx = 7 −4 4  c= 7  x 7−3  x 4 c
3
3
3

1
∫ sen2 x dx =∫ cos x sen x dx =∫ sec x tan x dx = sec x c
cos x
cos x

1

Ejercicio 1: Resolver lasintegrales indefinidas.
1
2

3
∫  x dx
∫ 12 dx

20 

33 

∫  x 1   3 x −2  dx
∫  2 t 2−1 2 dt
∫ y 2  y dy
∫  13 t  t2 dt
∫ dx
∫ 3 dt
∫  2 sen x  3 cos x  dx
∫  t 2− sen t  dt
∫  1−csc t cot t  dt
∫  2 sec2   d 
∫  sec2 − sen   d 
∫ sec y  tan y − sec y  dy
∫  tan2 y 1 dy

34 

∫

21 

x

∫ x 1 x dx

22 
23 
24 

5

∫ x x 23  dx
∫ 1 3 dx
2x

25 

6

∫

1
dx
2
3 x 

26 

7

∫  x 3  dx

8

∫  5− x  dx
∫  2 x −3 x 2  dx
∫  4 x 3 6 x 2−1  dx
∫  x 32  dx
∫  x 3− 4 x 2  dx

3
4

9
10 
11 
12 



3
2

13 

∫x

14 

∫  x 2 1 x


27 
28 
29 
30 
31 
32 

 2 x 1  dx





16 
17 

∫ 13 dx

18 

∫ 14 dx19 

∫

dx

3
∫  x 2 dx
4
∫   x 31  dx

15 

x 2 2 x − 3
∫ x 4 dx

x
x

x 2 x  1
dx
x

2

cos x
dx
1−cos 2 x

Integral definida.
La integral definida nos permite conocer el área existente bajo una función. Se representa
por:

b

∫a

f  x  dx

Ejemplos:
a

3

3

∣

∫1 4 dx = 4 x = 4  3− 4  1  = 12 −4 = 8 u 2
1

3

∣

[]

b

∫0  x  2  dx = 1 x 2 2 x = 1  3 2 2  3 − 1  0 2 2  0  = 9 6 = 21 u 2=10,5 u 2
2
2
2
2
2
0

c

∫0 sen x dx = −cos x

d

3





∣

= −cos  −[−cos  0  ] = 11 = 2 u 2

0

∫−1 ∣x∣ dx = −∫−1 x dx ∫0 x dx = − 1 x 2
2
1

0

1

0

∣

−1

1



∣

12
1
1
x = − [ 0 2−−1 2 ]  [ 12− 0 2 ]
2
2
2
0
11
2
=  = 1u22

3

e



∫1 −x 4 x −3  dx = − 1 x 3 4 x 2−3 x
3
2
3

2

3

∣ 

1
= − x 3  2 x 2− 3 x
3
1
=−

3

∣

1

[

 3 3
 1 3
 2  3 2−3  3 − −
 2  1 2 − 3  1 
3
3

]

1
1
4
= −9 18−9  − 2 3 = 1 = u2
3
3
3
0
f

g



∫1  x −3  dx = 1 x 3−3 x
3
2

2

4

4

∫1 3  x dx = 3∫1

2

∣

34
2

1
2

[3

]

3

2
1 
8
1
7
22
=
− 3  2 −
− 3  1  = − 6−  3 = − 3 = − u
3
3
3
3
3
3
1

x
x dx = 3
3
2

∣

= 2x

34
2

∣

4

∣

= 2  x 3 = 2  43− 2  13= 2  64− 2  1

1

1

1

= 2  8− 2  1  = 16 −2 = 14 u 2
h

i

 /4

∫0

sec 2 x dx = tan x

2


4

∣

1/ 2

= tan

0


−tan  0  = 1−0 = 1 u 2
4
2

∫0 ∣2...