1. UNIVERSIDAD NACIONAL DE HUANCAVELICA
ANALISIS MATEMATICO II
INTEGRANTES:
- ESPINOZA DAMIAN PABLO DIOSDADO
- HUIZA ANCCASI JHERIC MAX
- ARROYO MEJIA BEYKER
- INGA LAPA NELSON
- CICLO Y SECCIÓN:
- III “A”
2. 1) cos( )
, cos( )
, ( )
( ) ( cos( ))
( ) cos( )
( ) cos( )
x x dx
u x dv x dx
du dx v sen x
udv uv vdu
xsen x x
xsen x x
xsen x x C
2
2
2) cos( )
,
( )
cos( ) ,
1
,
1 1
( )
1
cos
cos( )
x bx dx
u x dv dx
sen bx
dv bx dx v
b
udv uv vdu
sen bx sen bx
x dx
b b
sen bx
x sen bx dx t bx
b b
senb x
x sen t dt
b b b
senb x
x bx
b b
xsen bx bx
C
b b
3.
2
2
2
2
3)
,
2
1
,
2 4
2 2
1 1
2 4 2 4
2 2
1 1
2
2 4 2 4
2 cos 2
1
2 4 4 8
1 (2 ) cos(2 )
4 4 8
xsen x dx
u x du dv
sen x
dv sen x dx v x
udv uv vdu
sen x sen x
x x x dx
x
sen x sen x
x x dx x
sen x x
x
x x
xsen x x
x C
2
2
4
2
2 2
2
4
2 2 2
4
3
2 2
4
2 2 4
2 2 4
4)
2
, '
1
' ,
2
2
2 2
1
2
2 2
1
1
2 1
1 1
1
2 2
1 1
1
2 2
xarcsen x dx
x
u arcsen x u
x
x
v x v
x x x
arcsen x dx
x
x arcsen x x x
dx
x
x
x arcsen x dx
x
x arcsen x x
x arcsen x x C
4.
2
2
2
2
5) (4 )
,
8
1
4 ,
2 16
8
1 1 (8 )
2 16 2 16
8
1 1 (8 )
2 16 2 16
1 (8 ) cos(8 )
2 16 4 128
1 (8 ) cos(8
4 16
xsen x dx
u x dv dx
sen x
dv sen x dx v x
udv uv vdu
sen x sen x
x x x dx
sen x sen x
x x dx dx
sen x x x
x x
xsen x
x
)
128
x
C
2
2
2
2
6)
2
1 1
cos 2 2
2 4
1 1
cos(2 ) 2
2 4
sen x dx
udv uv vdu
sen x x sen x xdx
sen x x x x sen x
xsen x x x sen x C
5.
2
2
1
2 2
2
7)
1
, '
1
' 1,
' '
1
1
1 1
2
1
2 1
2
1
arcsen u dx
u arcsen x u
x
v v x
uv uv u v
arcsen x x xdx
x
xarcsen x dv
u
xarcsen x x
xarcsen x x C
2
2 2
2
2
2
2
3
2 2
5 3
2 2 2
2 2
2
8) 1
1
1, '
2 1
' ,
2
' '
1
1
2 2
2 1
1
1
2 4 1
1 1
1 , 1
2 4 1
1 1 1
1 2
2 4
1 1 2 4
1 2
2 4 5 3
1 1
1 3 4 8 1
2 30
2 2
1 1
5 15
x x dx
u x u
x
x
v x v
uv uv u v
x x
x dx
x
x
x x dx
x
x
x x dx u x
x
x x u du udu du
u
x x u u u
x x x x x
x x x x
4
1
15
x C
6.
2
2
3
2
3 2
3 3
3 3
9) ln
ln
1
ln( ),
,
3
ln
3 3
1
ln
3 3 3
ln( )
3 9
x x dx
x x dx
u x du dx
x
x
dv x dx v
udv uv vdu
x x
x dx
x x
x
x x x
C
3
3
3
3 3
3 3
3 3
10)
,
cos
, cos
3
cos cos
cos cos
3 3
cos cos
cos cos
3 3
cos
cos
3 3 4
cos
cos
xsen x dx
u x du dx
x
dv sen x dx v x
udv uv vdu
x x
x x x dx
x x
x x x dx
x sen x sen x
x x sen x
x x
x x
3 3
2
3 4
sen x sen x
C
7.
3 2
3 2
3
2
2 2
3 2
2
3 2 2
2
3 2 2
2 2 2
3 2
2 2 2 2
3 2
2 2 2
3 2
1
, 3
,
2
3
2 2
1
3
2 2
3
2 2
3
2
2 2 2 2
3
2 2 2 2 2
3 1 1
2 2 2 2 2 2
x
x
x
x x
x
x
x
x
x x x
x x x x
x x x
x e dx
u x du x dx
e
dv e dx v
udv uv vdu
e e
x x dx
e
x e x dx
e
x x e dx
e e e
x x xdx
e e e e
x x x dx
e e e
x x x
23
3 2 2 2 23 2
3 3 3
2 4 8
x x x
e
x e x e xe e
C
2
2
2
2
2 2
2
2
2 2 2
2 2
2
2
2 cos
cos
cos( ),
,
2
cos
2 2
1
cos( )
2 2
1
cos( ) cos
2 2 2 2
1 1
cos cos cos
2 2 2 2
co
x
x
x
x
x x
x
x
x x x
x x
x x
x
e x dx
x e dx
u x du sen x dx
e
dv e dx v
e e
x sen x dx
e
x e sen x dx
e e e
x sen x x dx
e e
e x dx x sen x e x dx
e
2 2
2
2 2
2
2 2
cos 1
s cos
2 4 4
cos
5
cos
4 2 4
2 cos
5
x x
x
x x
x
x x
x e sen x e
x dx e x dx
x e sen x e
e x dx
e x sen x e
C
8.
3
3
4
3
4 4
4 3
4
3
4 4
4 4
3 ln
ln
1
ln ,
,
4
1
ln
4 4
ln
4 4
1
ln
4 4
1
ln
4 4 4
ln
4 16
x x dx
x x dx
u x du dx
x
x
dv x dx v
x x
x dx
x
x x
x dx
x
x x dx
x x
x
x x x
C
2
2
2
2
2
2
4
1
arctan ,
1
1 ,
1
arctan
1
arctan , 1
1
1
arctan
2
1 1
arctan
2
1
arctan ln
2
1
arctan ln 1
2
1
arctan ln 1
2
arctg x dx
u x du dx
x
dv dx v x
x x x dx
x
x
x x dx t x
x
x x dt
t
x x dt
t
x x t
x x x
x x x C
9.
3
4
3
4 4
4
4
4
4
4 3 2
4 3 2
4
4 3
5 ln 3
1
ln 3 , '
3
' ,
4
1
ln 3
4 3 4
1
ln 3
4 4 3
1 1
ln 3
4 4 3
1 1 81
ln 3 3 9 27
4 4 3
1 1 81
ln 3 3 9 27
4 4 3
1 1 9
ln 3
4 4 4
x x dx
u x u
x
x
v x v
x x
x dx
x
x
x x dx
x
x
x x dx
x
x x x x x dx
x
x x x dx x dx xdx dx dx
x
x
x x x
2
27 81ln 3
2
x
x x C
3
3
3
3 3
3
3
3
3
3
3
3 3
) 2 1
2 1, 2
,
3
2 1 2
3 3
1
2 1 2
3 3
1
2 1 2
3
2 1
2 1
3 3 3
6
9
x
x
x
x x
x
x
x
x
x
x
x x
a x e dx
u x du dx
e
dv e dx v
e e
x dx
e
x e dx
e
x e dx
e
e
x e
xe e
C
10.
3 2
3 2
2
2
2 2
3 2
2
3 2 2
2 2 2
3 2
2 2 2
3 2
)
, 3
,
2
3
2 2
1
3
2 2
3
2
2 2 2 2
3
2 2 2 2
x
x
x
x x
x
x
x x x
x x x
b x e dx
u x dv x dx
e
dv e dx v
e e
x x dx
e
x e x dx
e e e
x x xdx
e e e e
x x x
2
2 2 2
3 2 2
2
3
2 2 2 2
2
3
2 2 2
3 2
2 2 2
2
3 1 1
2 2 2 2 2 2
1 3 1
2 2 2 2 4
3 3
2 4 8
3 3 3
2 4 8
x
x x x
x
x x x x
x x x
x x x
dx
e e e
x x x x e
x x
x
e e e e
x x
x
e e e
x x x
C
x e e
5
2
2
2
)
,
5
5 ,
ln 5
5 5
ln 5 ln
5 1
5
ln 5 ln 5
5 1 5
ln 5 ln 5 ln 5
5 5
ln 5 ln 5
5 5
ln 5 ln 5
x
x
x x
x
x x
x x
x
c x dx
u x du dx
dv dx v
x dx
x
x dx
x
x
x
C
11.
2
2
2
2
2
2
2
) ln
1
ln , 2ln
1 ,
1
ln 2ln
1
ln 2 ln
1
ln 2 ln
ln 2 ln
ln 2 ln 2
d x dx
u x du x dx
x
dv dx v x
x x x x dx
x
x x x x dx
x
x x x x x dx
x
x x x x x
x x x x x C
3
3
3
3 2
3 2
3 2
3 2
3
) 2
2
2
)
16
1
16
1
cos cos 3
16
1
cos 3 2
16
1
cos 3 2 cos
16
1
2 cos 2 3 2 2 2 2 cos 2 2
16
cos
e x sen x dx
t
x t x
t sen t
i dx
axf x dx ax f x dx
t sen t dt
udv uv vdu
t t t t dt
t t t sen t tsen t dt
t t t sen t t t sen t
x x x sen x x x sen x
x
2
2 3 2 3 cos 2 3 2
2 4 8
x x sen x x x sen x
C
12.
2
2
) sec 3
) ,
tan 3
3 ,
3
tan 3 tan 3
)
3 3
tan 3 1 1
tan
3 3 3
tan 3 1 1
3 9
tan 3 1
ln cos
3 9
tan 3 1
ln cos 3
3 9
f x x dx
udv uv vdu
i u x du dx
x
dv sex x dx v
x x x
ii udv dx
x x
t dt
x x
du
u
x x
t
x x
x C
2 2
2 2
2 2
) ln 2
) ln 2ln
( ) ( ) ( )
ln 2ln
ln
2 ln 2
2 4
2
ln 2 2 2
2 2 ln 2 2 2
2 4
ln 2 4 4 4 12
2 2 ln 2
2 4
g x x dx
i t t dt t dt
f x g x dx f x dx g x dx
t t dt t dt
t t t
t t t
t x
x x x
x x x
x x x x x
x x C
13.
2
2
) csc cot
cos
1
) .
cos
1 1
1
1 cos
1 1 1
ln
2 1
cos
cos 1
1 1
ln
2 cos 1
ln tan
2
h x x x dx
x
i x dx
sen x sen x
x x
dx
sen x
x dx
sen x sen x
senx
x dx
sen x x
t
x
sen x t
t x
x
x
sen x x
x x
sen x
C
2
2
)
sec tan
) tan tan
tan
cos
1
tan
1
tan
tan ln
cos
tan ln cos
i xsen x dx
u x du dx
dv x dx v x
i x x x dx
sen x
x x dx
x
x x dt
t
x x dt
t
x x t
t x
x x x C
14.
2
2
2 2
2
2 2
2
2
2
2 2
2 2
2
2
) arctan
1
arctan
1
2
1
)arctan
2 1
arctan
2 1
1 1 1
arctan
2 1
1 1 1
arctan
2 1 1
1
arctan arctan
2
arctan arctan
2 2
j x x dx
u x du dx
x
x
dv xdx v
x x
i x dx
x x
x x
x dx
x x
x x
x dx
x x
x x
x dx dx
x x x
x
x x x
x
x x x x
C
2
2
2
2
2
2
) 3arccos
1
cos
1
3 3
1
)arccos 3 3
1
1
arccos 3 1 3
1
arccos 3 3
1
arccos 3 3 1
arccos 3 3 1
1
arccos 3 3 1
k x dx
u ar x du dx
x
dv v x
i x x x dx
x
x x x dx
x
x
x x dx
x
x x dt
x x t
t x
x x x C
15.
2
2
2
2
) arccot 2
2
arccot
)
2
1
arccot
2
1 1
arccot
2 1
1
arccot
2 1
1 1
arccot
2 2
1 1
arccot ln
2 2
1
1 1
arccot ln 1
2 2
2
1 1
arccot 2 2 ln
2 2
l x dx
t x
t
i dt
t dt
t t t dt
t
t
t t dt
t
t t du
u
t t u
u t
t t t
t x
x x
2
2
1 2
1
arccot 2 ln 1 4
4
x
x x x C
2
2
2
2
2 2
2 2
2
2
2
3
2
2
3 2
)
2
)
1
2
)
2
2 2 1
2
2 1
)
1
1
2
2
)
1 1
2 2
1
)
1 1
2
m xarcsen x dx
x
i u arcsenx du
x
x
du x v
ii
x x x
xarcsenx dx arcsenx dx
x
x x
dx
x
iii
x
dx
x
u x
dv du
x dx
dx x
iv
x du x u
du du
x
u u u
u
du A
u
v
A
u
3
1
2
2
1 3
2 2
2 1 3
2
2 2
1 2
2
2 3
2
1 1
3
2
1 1
2 3
u
du
u
u
A u
A x x
x
arcsenx x x C
16.
1 1
2 2
3 1
2 2
3 1
2 2
2
2
) 1
1
)
.
2 2
5 3
1
2 1 1 2 1 1
5 3
2 1 2 1 2 1 1
5 3
n x x dx
t x
i t t tdt
t t t dt
t t dt
t dt t dt
t t t t
t x
x x x x
x x x x x
C
1
2
1 1
2 2
1
2
1
2
)
5 2
5 2
5
)
4
1 5
4
1 5
4
1 4
4
1 5
4
1 2
10
4 3
5 2
2 5 2 5 2
1
10 5 2
4 3
5 2 5 2 5
5 2
6 2
x
o dx
x
t x
t
i dt
t
t
dt
t
t
dt
t
t
dt
t t
t dt dt
t
t t
t
t x
x x
x
x x
x C
17.
2
2
2 2
2 2
2
2
2 2
2 2
2
2
) 3
) 3
3 cos 3 3
2 2
3 1
3 cos 3 1 3 3
2 2 2 2
3 3
......
3 3
3 3 cos 3 3
2 2 2 2
3 3cos 3
3
2
x
x
x x
x x
x
x
x x
x x
x
x
p e sen x dx
i sen x e dx
e e
sen x x dx
e e
sen x x e sen x dx
e sen x dx
e e
e sen x dx sen x x e sen x dx
sen x e x
e sen x dx
2
2
2 2
2 2
2 2
2
2 2
2
2 2
9
3
4 4
3 3cos 3
9
3 3
4 2 4
3 3cos 3
13
3
4 2 4
2 3 3cos 3
3
13 13
2 3 3 cos 3
13
x
x
x x
x x
x x
x
x x
x
x x
e
e sen x dx
sen x e x e
e sen x dx e sen x dx
sen x e x e
e sen x dx
sen x e x e
e sen x dx
e sen x e x
C
5
5
5 5
5
5
5 5
5
5
5 5
5 5
5
) cos 2
) cos 2
cos 2 2 2
5 5
2
cos 2 2
5 5
2 1
cos 2 2 2 cos 2
5 5 5 5
cos 2
...........
2 2
cos 2 cos 2 2 cos 2
5 5 5 5
cos 2
x
x
x x
x
x
x x
x
x
x x
s x
s
q e x dx
i x e dx
e e
x sen x dx
e
x sen x e dx
e e
x sen x e x dx
e x dx
e e
e x dx x sen x e x dx
e
5 5
5
5 5
5
5 5
5
cos 2 2 2 4
cos 2
5 25 25
cos 2 2 2
29
cos 2
25 5 25
5cos 2 2 2
cos 2
29
x x
x
x x
s
x x
s
x e sen x e
x dx e x dx
x e sen x e
e x dx
x e sen x e
e x dx C
18.
1
2
0
2
2
2
2 2
2 1
0
2
2 2 3
1
0
2 16
3
2 2 1
0
2
2 2 2 2 1
0
2 2 2
) ln 16
2
ln 16 '
16
'
2
2
ln 16
2 16 2
ln 16
2
1
ln 16
2 16
1 1
ln 16 16 16ln 16
2 2
1
ln 16 16 16ln 16
2
r x x dx
x
u x u
x
x
v x v
x x x
x dx
x
x x x
dx
x
x
x x dx
x
x x x x
x x x
2 1
0
17ln 17 1 64ln 2
2
x
) ln
) ln
cos
cos
cos
cos
cos
2 cos
t
t t
t t t
t t t
t
t t t t
t t t t
t t t
t
s sen x dx
i t x
sen t e dt
sen t e e t dt
sen t e t e e sen t dt
sen t e t e e sen t dt
e sen t dt
e sen t dt sen t e t e e sen t dt
e sen t dt sen t e t e e sen t dt
e sen t dt sen t e t e
e sen t dt
ln ln
cos
2 2
) ln
ln cos ln
2 2
ln cos ln
2
t t
x x
sen t e t e
ii t x
sen x e x e
sen x x x x
C
19.
)
2
2
2 cos cos
2 cos
2 cos
2 cos 2
t sen x dx
t x
tsen t dt
tsen t dt
t t t dt
t t sen t
Sustituyendot x
x x sen x
x x sen x C
2
2
2
2
2
2
2
2
2
2
2
2
2
2 2
2
2
2 2
2
2
) ln 1
) ln 1
1
ln 1 2
1
1
1
) ln 1 2
1
ln 1 2
1
1 1
ln 1 2
1
1 1
ln 1 2
1 1
1 1
ln 1 2
1 1
ln 1 2 arctan
ln 1
u x dx
i x dx
u x du xdx
x
dv dx v x
ii x x x xdx
x
x
x x dx
x
x
x x dx
x
x
x x dx
x x
x
x x dx dx
x x
x x x x
x
2 2arctan
x x x C
20. 2
2 2
)
) 2
)
2
x
t
t
t
t t
x x
v e dx
i t x
te dt
te e dt
te e
ii Sustituyendo
xe e C