1. Universidad autónoma de ciudad
Juárez (instituto de ingeniería y
tecnología)
Maestro: Carlos López Rubalcaba
Tarea: 2
Alumno: Salvador Gómez Villa
Matricula: 132096
Materia: Calculo II
Grupo:

2. Integral de la potencia de una suma.
( )
2
1 3
2: ( ) ?
v a bx,n 2,dv bdx
( )
1
1 3
n
n
a bx b dx
v a bx
v dv c
n
+
− − =
= − = = −
−
= + = +
+
∫
∫
2 2
2 2 2
1 2 2 1
2
6
4: ?
(5 3 )
(5 3 ) 6 , 5 3 , 2, 6
(5 3 ) 1
1 2 1 (5 3 )
n
n
xdx
x
x xdx v x n dv xdx
v x
v dv c c
n x
−
+ − +
=
+
= + = + = − =
+
= = + = = − +
+ − + +
∫
∫
∫

3. 1
2
1
2
1
2
2
2 2 1
2
1 2
2
1
2
2
6: ?
( )
( ) 2 , ( ),n , 2
( )
2
1
n
n
dx
a x
a x dx v a x dv dx
v a x
v dv c a x c
n
−
+
−
=
−
= − − = − = − = −
−
= = + = = − +
+
∫
∫
∫
1
2 1
2
1 3
1 31 1 12 2
2 2
31
2 2
8: ?
2
1
( ) , ( ),n ,
2
( )( ) ( )
1 1 3
n
n
a bt
bdt
a bt bdt v a bt dv bdt
a bta bt a btv
v dv c c
n
+
+
+
=
= + = + = =
++ +
= = + = = = +
+ +
∫
∫
∫
1
2
1
2
2
2 2 1
22
121 21
4
1
2
(2 3)
10: ?
4 3
1 (2 3)
( 3 ) (2 3) , ( 3 ), , (2 3)
4 3
( 3 ) 3
,
1 1 2
n
n
s ds
s s
s ds
s s s ds v s s n dv s ds
s s
s sv s s
v dv c c
n
−
− ++
+
=
+
+
= = + + = + = − = +
+
+ +
= = + = +
+ − +
∫
∫ ∫
∫
Casos especiales:

4. 2 3
2 3 2 3
2 3 2 3
2 3 11
2 428
(7 1)
2: ?
2
1
(7 1) , (7 1) , 3, 14
2
1 1 1
(7 1) 14 (7 1) 14
2 14 28
(7 1) 1
(7 1)
4 112
x
xdx
x xdx v x n dv xdx
x xdx x dx
x
x c
+
−
=
= − = − = = =
= − = − =
−
= = − +
∫
∫
∫ ∫
2
2 2
2 1 11
2 2 22
4: (2 ) ?
(2 ) , (2 ), 1, 2
(2 )1 1
(2 )2 (2 )
2 2 4
x x dx
x xdx v x n dv xdx
x
x xdx x c
+
+ =
= + = + = =
+
= + = = + +
∫
∫
∫
1
2
1
2
31
2 2
2
2 2 1
2
121
2 24
3
2
10: 3 2 ?
(3 2 ) , 3 2 , , 4
(3 2 )1 1
(3 2 ) (4) (3 2 )
4 6
u u du
u udu v u n dv udu
u
u udu u c
+
− =
= − = − = =
−
= − = = − +
∫
∫
∫

5. 3 5 2
3 2
3 5 11
3 5 2 3 63
6: ( 2) ?
( 2),n 5,dv 3
( 2)1 1
( 2) (3) ( 2)
3 6 18
x x dx
v x x dx
x
x x dx x c
+
+ =
= + = =
+
= + = = + +
∫
∫
2 2
2 2 2
2 2
2 2 13
2 2 2
2
3
12: ?
( 3)
3 3 ( 3) , ( 3),n 2,dv 2xdx
( 3)
( 3)1 3
3 ( 3) (2)
2 1 2( 3)
xdx
x
xdx
x xdx v x
x
x
x xdx c
x
−
− +
−
=
+
= + = + = − =
+
+
+ = = − +
− +
∫
∫ ∫
∫
Integral de las funciones exponenciales.
2
2
1
2
1
2 2
2:8 ?
, 8,
8
ln ln8
x
x
x
u
u
dx
v a dv dx
a
a du c c
a
=
= = =
= + = +∫

6. 5
5
4: 3 5 ?
5 , 3 , 5
3
ln ln3
x
u x
u
a dx
v x a a dv dx
a a
a du c c
a a
− =
= = − =
−
= + = +
∫
∫
4
4
4
4 4
10: ?
7
7 , 4 , 7, 4
1
7 (4) ,
4 ln
1 7 7
4 ln 7 4(ln 7)
x
x
u
x u
x x
dx
dx v x a dv dx
a
dx a du c
a
c c
−
−
− −
=
= = − = = −
= − = + =
+ = +
∫
∫
∫ ∫
2
2
2
2 2
12: 5 ?
5 , 2 , , 2
1
5 (2)
2 ln
5 5
2 lne 2ln 2
t
t
u
t u
t t
e dt
e dt v t a e dv dt
a
e dt a du c
a
e e
c
=
= = = =
= = = +
= = +
∫
∫
∫ ∫

7. 4
4
4 4
4
14: 3 ?
3 , 4 , , 4
1 3 3
3( ) (4) ( )
4 4 ln 6 4ln6
ln
r
r
r r
r
u
u
e dr
e dr v r a e dv dr
e e
e dr c
a
a du c
a
=
= = = =
= = = +
= +
∫
∫
∫
∫

8. Integrales en que intervienen la tangente, cotangente, secante y
cosecante.
2
1
2 2
2
2
2: ?
,
1
2 ( )
2
ln cos 2ln cos
x
x
x
x
tg dx
v dv dx
tg dx
v c c
=
= =
− + = − +
∫
∫
4: 3 2 ?
3 2 , 2 ,dv 2dx
1 3
3( ) 2 (2) ln 2
2 2
ln
ctg xdx
ctg xdx v x
ctg x dx sen x c
senv c
=
= = =
= = +
+
∫
∫
∫
1 1 1
2 2 2
1 1
2 2
6: ?
1
( ) ( ) dx, ( )
2
1
2 ( ) ( ) ( )
2
ln 2ln
ctg x
dx
x
x ctg x v x dv x
ctg x x dx
senv c sen x c
−
−
=
= = =
=
+ = +
∫
∫
∫

9. 2
2
2 2 2
8: sec ?
3
2
,
3 3
3 2 3
sec ( ) ln sec
2 3 3 2 3 3
ln sec
x
xdx
x
v dv xdx
x x x
xdx tg c
v tgv c
=
= =
= = + +
+ +
∫
∫
3 2
3 2
3 2 3
10: ?
, 3 , ln cos
1 1
(3) cos(x)
3 3
tgx x dx
v x dv x tgdv v c
tgx x dx c
=
= = = − +
= = − +
∫
∫
∫
Segundo caso:
2
2: ?
3 cos2
1 2
, 3 cos2 , 2
3 cos2
1
ln 3 cos2
3
sen xdx
x
sen xdx
v x dv sen xdx
x
dv
v c x c
v
=
+
−
= − = + =
= + = − + +
∫
∫
∫

10. 2
2
csc
4: ?,
3
3 , csc
ln ln 3
u
du
ctgu
v ctgu dv udu
dv
v c ctgu c
v
=
−
= − =
= = + = − +
∫
∫
Tercer caso:
1
2: ?
( )
1
( ) , ( )
( ( )
ln cos( )
b
dt
ctg a bt
b b
dt tg dy tg a bt b v a bt dv b
ctg a bt a bt
a bt c
−
=
−
= = = − = − = −
− −
= − − +
∫
∫ ∫ ∫
Integrales que conducen a las funciones trigonometricas.
2: cos ?
2
1
,
2 2
1
2 cos ( ) 2
2 2 2
x
dx
x
v dv
x x
dx sen c
=
= =
= +
∫
∫

11. 2
2
2 2
4: cos(1 ) ?
(1 ),dv 2x
1 1
(1 )(2) (1 )
2 2
x xdx
v x
x xdx sen x c
− =
= − = −
− − = − − +
∫
∫
2
6: ( ) ?
3 2
2 1
( ) , ( ),
3 2 2 2
2 1 1 1
( ) ( )( ) cos(a )
3 2 2 2 6 2
x
sen a dx
x x
sen a dx v a dv
x x
sen a dx c
− =
= − = − = −
= − − − = − +
∫
∫
∫
2
14: ?
3csc3
2
, 3 , 3
3 csc3
2
csc3u c
3
du
u
du
v u dv du
u
=
= = =
= +
∫
∫

12. Casos especiales.
2 2 2
2 2 2 2
2 2
2
2
2: ?
1 cos2
2 1 cos2 2(1 cos2 ) 1 cos2
2
1 cos2 1 cos2 1 cos 2 2
1 cos2 1
2 csc 2 cos2 ( 2 ) 2csc 2 cos2 ( 2 )
2 2 2
1
2
dx
x
dx x x x
dx
x x x sen x
x
dx xdx x sen x xdx x sen x
sen x sen x
ctg x c
sen x
− −
=
−
+ + +
= = = =
− + −
= + = + = +
= − − +
∫
∫ ∫ ∫
∫ ∫ ∫ ∫ ∫ ∫