Solucion de problemas de ecuaciones difrenciales hasta 19JAVIERTELLOCAMPOS
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Solucion de problemas de ecuaciones difrenciales hasta 19JAVIERTELLOCAMPOS
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Solid waste management & Types of Basic civil Engineering notes by DJ Sir.pptxDenish Jangid
Solid waste management & Types of Basic civil Engineering notes by DJ Sir
Types of SWM
Liquid wastes
Gaseous wastes
Solid wastes.
CLASSIFICATION OF SOLID WASTE:
Based on their sources of origin
Based on physical nature
SYSTEMS FOR SOLID WASTE MANAGEMENT:
METHODS FOR DISPOSAL OF THE SOLID WASTE:
OPEN DUMPS:
LANDFILLS:
Sanitary landfills
COMPOSTING
Different stages of composting
VERMICOMPOSTING:
Vermicomposting process:
Encapsulation:
Incineration
MANAGEMENT OF SOLID WASTE:
Refuse
Reuse
Recycle
Reduce
FACTORS AFFECTING SOLID WASTE MANAGEMENT:
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
This presentation provides an introduction to quantitative trait loci (QTL) analysis and marker-assisted selection (MAS) in plant breeding. The presentation begins by explaining the type of quantitative traits. The process of QTL analysis, including the use of molecular genetic markers and statistical methods, is discussed. Practical examples demonstrating the power of MAS are provided, such as its use in improving crop traits in plant breeding programs. Overall, this presentation offers a comprehensive overview of these important genomics-based approaches that are transforming modern agriculture.
Power-sharing Class 10 is a vital aspect of democratic governance. It refers to the distribution of power among different organs of government, levels of government, and social groups. This ensures that no single entity can control all aspects of governance, promoting stability and unity in a diverse society.
For more information, visit-www.vavaclasses.com
Basic Civil Engineering Notes of Chapter-6, Topic- Ecosystem, Biodiversity Green house effect & Hydrological cycle
Types of Ecosystem
(1) Natural Ecosystem
(2) Artificial Ecosystem
component of ecosystem
Biotic Components
Abiotic Components
Producers
Consumers
Decomposers
Functions of Ecosystem
Types of Biodiversity
Genetic Biodiversity
Species Biodiversity
Ecological Biodiversity
Importance of Biodiversity
Hydrological Cycle
Green House Effect
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
1. FORMULARIO DE
CÁLCULO DIFERENCIAL
E INTEGRAL VER.3.6
Jesús Rubí Miranda (jesusrubi1@yahoo.com)
http://mx.geocities.com/estadisticapapers/
http://mx.geocities.com/dicalculus/
VALOR ABSOLUTO
1 1
1 1
si 0
si 0
y
0 y 0 0
ó
ó
n n
k k
k k
n n
k k
k k
a a
a
a a
a a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥
=
− <
= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
( )
( )
/
p q p q
p
p q
q
q
p pq
p p p
p p
p
q
p q p
a a a
a
a
a
a a
a b a b
a a
b b
a a
+
−
⋅ =
=
=
⋅ = ⋅
=
=
EXPONENTES
10
log
log log log
log log log
log log
log ln
log
log ln
log log y log ln
x
a
a a a
a a a
r
a a
b
a
b
e
N x a N
MN M N
M
M N
N
N r N
N N
N
a a
N N N N
= ⇒ =
= +
= −
=
= =
= =
LOGARITMOS
ALGUNOS PRODUCTOS
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
2 2
2 2 2
2 2 2
2
2
3 3 2 2 3
3 3 2 2 3
2 2 2
2
2
3 3
3 3
a c d ac ad
a b a b a b
a b a b a b a ab b
a b a b a b a ab b
x b x d x b d x bd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
a b a a b ab b
a b a a b ab b
a b c a b c
⋅ + = +
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + 2
2 2 2
ab ac bc
+ + +
( ) ( )
( ) ( )
( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5
1
1
n
n k k n n
k
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a b a b n
− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
− ⋅ = − ∀ ∈
∑
5
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
5 4 3 2 2 3 4 5 6 6
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a a b a b a b ab b a b
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
CA
CO
HIP
θ
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x : ( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1
sen sen 2sen cos
2 2
1 1
sen sen 2sen cos
2 2
1 1
cos cos 2cos cos
2 2
1 1
cos cos 2sen sen
2 2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
-5 0 5
-2
-1
0
1
2
3
4
arc ctg x
arc sec x
arc csc x
( ) ( )
( ) ( )
1 1
1
1 1
1
1 impar
1 par
n
k n k k n n
k
n
k n k k n n
k
a b a b a b n
a b a b a b n
+ − −
=
+ − −
=
+ ⋅ − = + ∀ ∈
+ ⋅ − = − ∀ ∈
∑
∑
( )
( )
1 2
1
1
1 1
1 1 1
1 0
1
n
n k
k
n
k
n n
k k
k k
n n n
k k k k
k k k
n
k k n
k
a a a a
c nc
ca c a
a b a b
a a a a
=
=
= =
= = =
−
=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
SUMAS Y PRODUCTOS
n
θ sen cos tg ctg sec csc
0 1 0 ∞ 1 ∞
1 2 3 2 1 3 3 2 3 2
1 2 1 2 1 1 2 2
60 3 2 1 2 3 1 3 2 2 3
90 0 1
∞ 0 ∞
0 ( )
sen
tg tg
cos cos
α β
α β
α β
±
± =
⋅
30
45
( ) ( )
( ) ( )
( ) ( )
1
sen cos sen sen
2
1
sen sen cos cos
2
1
cos cos cos cos
2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +
⋅ = − − +
⋅ = − + +
2
2
2 2
2 2
sen cos 1
1 ctg csc
tg 1 sec
θ θ
θ θ
θ θ
+ =
+ =
+ =
IDENTIDADES TRIGONOMÉTRICAS
1
[ ]
[ ]
sen ,
2 2
cos 0,
tg ,
2 2
1
ctg tg 0,
1
sec cos 0,
1
csc sen ,
2 2
y x y
y x y
y x y
y x y
x
y x y
x
y x y
x
π π
π
π π
π
π
π π
= ∠ ∈ −
= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
= ∠ = ∠ ∈ −
tg tg
tg tg
ctg ctg
α β
α β
α β
+
⋅ =
+
( )
( )
( )
sen sen
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
senh
2
cosh
2
senh
tgh
cosh
1
ctgh
tgh
1 2
sech
cosh
1 2
csch
senh
x x
x x
x x
x x
x x
x x
x x
x x
e e
x
e e
x
x e e
x
x e e
e e
x
x e e
x
x e e
x
x e e
−
−
−
−
−
−
−
−
−
=
+
=
−
= =
+
+
= =
−
= =
+
= =
−
FUNCIONES HIPERBÓLICAS
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
sen 2 sen
cos 2 cos
tg 2 tg
sen sen
cos cos
tg tg
sen 1 sen
cos 1 cos
tg tg
n
n
n
n
n
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( ) ( )
( )
( )
( )
( )
( )
( )
1
1
1
2
1
2 3 2
1
3 4 3 2
1
4 5 4 3
1
2
1
1 2 1
2
=
2
1
1 1
1
2
1
2 3
6
1
2
4
1
6 15 10
30
1 3 5 2 1
!
k
n
n
k
k
n
k
n
k
n
k
n
k
n
k
n
a k d a n d
n
a l
r a rl
ar a
r r
k n n
k n n n
k n n n
k n n n n
n n
n k
n n
k
=
−
=
=
=
=
=
=
+ − = + −
+
− −
= =
− −
= +
= + +
= + +
= + + −
+ + + + − =
=
=
∑
∑
∑
∑
∑
∑
∏
( )
( )
0
!
,
! !
n
n n k k
k
k n
n k k
n
x y x y
k
−
=
≤
−
+ =
∑
Gráfica 1. Las funciones trigonométricas: ,
cos x , tg x :
sen x
-8 -6 -4 -2 0 2 4 6 8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sen x
cos x
tg x
[
{ }
]
{ } { }
senh :
cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
( )
( ) ( )
( )
( )
sen 0
cos 1
tg 0
2 1
sen 1
2
2 1
cos 0
2
2 1
tg
2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+
= −
+
=
+
= ∞
Gráfica 2. Las funciones trigonométricas ,
sec x , ctg x :
csc x
Gráfica 5. Las funciones hiperbólicas ,
, :
senh x
cosh x tgh x
( ) 1 2
1 2 1 2
1 2
!
! ! !
k
n n
n n
k k
k
n
x x x x x x
n n n
+ + + = ⋅
∑
-8 -6 -4 -2 0 2 4 6 8
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
csc x
sec x
ctg x
-5 0 5
-4
-3
-2
-1
0
1
2
3
4
5
senh x
cosh x
tgh x
sen cos
2
cos sen
2
π
θ θ
π
θ θ
= −
= +
( )
3.14159265359
2.71828182846
e
π =
=
CONSTANTES
…
…
( )
( )
( )
( )
2 2
2
2
2
2
sen sen cos cos sen
cos cos cos sen sen
tg tg
tg
1 tg tg
sen 2 2sen cos
cos2 cos sen
2tg
tg 2
1 tg
1
sen 1 cos2
2
1
cos 1 cos2
2
1 cos2
tg
1 cos2
α β α β α β
α β α β α β
α β
α β
α β
θ θ θ
θ θ θ
θ
θ
θ
θ θ
θ θ
θ
θ
θ
± = ±
± =
±
± =
=
= −
=
−
= −
= +
−
=
+
∓
∓
1
sen csc
sen
1
cos sec
cos
sen 1
tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θ θ
θ
θ θ
θ
θ
θ θ
θ θ
= =
= =
= = =
TRIGONOMETRÍA
Gráfica 3. Las funciones trigonométricas inversas
arcsen x , arccos x , arctg x :
-3 -2 -1 0 1 2 3
-2
-1
0
1
2
3
4
arc sen x
arc cos x
arc tg x
radianes=180
π
( )
( )
1 2
1 2
1
1
2
1
2
1
senh ln 1 ,
cosh ln 1 , 1
1 1
tgh ln , 1
2 1
1 1
ctgh ln , 1
2 1
1 1
sech ln , 0 1
1 1
csch ln , 0
x x x x
x x x x
x
x x
x
x
x x
x
x
x x
x
x
x x
x x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+
= <
−
+
= >
−
± −
= < ≤
+
= + ≠
FUNCS HIPERBÓLICAS INVERSAS
2. ( ) ( ) ( )( )
( )( )
( )
( )( )
( ) ( ) ( )
( )
( )
( )
( )
( )
2
0 0
0 0 0
0 0
2
2 3
3 5 7 2 1
1
''
'
2!
: Taylor
!
'' 0
0 ' 0
2!
0
: Maclaurin
!
1
2! 3! !
sen 1
3! 5! 7! 2 1 !
cos
n
n
n n
n
x
n
n
f x x x
f x f x f x x x
f x x x
n
f x
f x f f x
f x
n
x x x
e x
n
x x x x
x x
n
x
−
−
−
= + − +
−
+ +
= + +
+ +
= + + + + + +
= − + − + + −
−
=
ALGUNAS SERIES
( )
( )
( ) ( )
( )
2 4 6 2 2
1
2 3 4
1
3 5 7 2 1
1
1 1
2! 4! 6! 2 2 !
ln 1 1
2 3 4
tg 1
3 5 7 2 1
n
n
n
n
n
n
x x x x
n
x x x x
x x
n
x x x x
x x
n
−
−
−
−
−
− + − + + −
−
+ = − + − + + −
∠ = − + − + + −
−
( )
( )
2 2
2 2
sen cos
sen
cos sen
cos
au
au
au
au
e a bu b bu
e bu du
a b
e a bu b bu
e bu du
a b
−
=
+
+
=
+
∫
∫
MAS INTEGRALES
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
2
2 2 2 2
2
2 2 2 2 2 2
sen
cos
ln
1
ln
1
cos
1
sec
sen
2 2
ln
2 2
du u
a
a u
u
a
du
u u a
u a
du u
a
u a u a a u
du a
a u
u u a
u
a a
u a u
a u du a u
a
u a
u a du u a u u a
= ∠
−
= −∠
= + ±
±
=
± + ±
= ∠
−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
INTEGRALES CON
( )
( )
2 2
2 2
2 2
2 2
2 2
1
tg
1
ctg
1
ln
2
1
ln
2
du u
a a
u a
u
a a
du u a
u a
a u a
u a
du a u
u a
a a u
a u
= ∠
+
= − ∠
−
= >
+
−
+
= <
−
−
∫
∫
∫
INTREGRALES DE FRAC
( )
( )
1
tgh ln cosh
ctgh ln senh
sech tg senh
csch ctgh cosh
1
ln tgh
2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫
∫
∫
∫
( )
( )
( )
2 2
2 2
2
cosh senh 1
1 tgh sech
ctgh 1 csch
senh senh
cosh cosh
tgh tgh
x x
x x
x x
x x
x x
x x
− =
− =
− =
− = −
− =
− = −
IDENTIDADES DE FUNCS HIP
( )
( )
( )
2 2
2
senh senh cosh cosh senh
cosh cosh cosh senh senh
tgh tgh
tgh
1 tgh tgh
senh 2 2senh cosh
cosh 2 cosh senh
2tgh
tgh 2
1 tgh
x y x y x y
x y x y x y
x y
x y
x y
x x x
x x x
x
x
x
± = ±
± = ±
±
± =
±
=
= +
=
+
( )
( )
2
2
2
1
senh cosh 2 1
2
1
cosh cosh 2 1
2
cosh 2 1
tgh
cosh 2 1
x x
x x
x
x
x
= −
= +
−
=
+
senh 2
tgh
cosh 2 1
x
x
x
=
+
2
2
2
0
4
2
4 discriminante
ax bx c
b b ac
x
a
b ac
+ + =
− ± −
⇒ =
− =
OTRAS
( )
1
0
0
0
0
1
lim 1 2.71828...
1
lim 1
sen
lim 1
1 cos
lim 0
1
lim 1
1
lim 1
ln
x
x
x
x
x
x
x
x
x
x e
e
x
x
x
x
x
e
x
x
x
→
→∞
→
→
→
→
+ = =
+ =
=
−
=
−
=
−
=
LÍMITES
( )
( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x
x x
n n
f x x f x
df y
D f x
dx x x
d
c
dx
d
cx c
dx
d
cx ncx
dx
d du dv dw
u v w
dx dx dx dx
d du
cu c
dx dx
∆ → ∆ →
−
+ ∆ − ∆
= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
DERIVADAS
( )
( )
( ) ( )
( )
2
1
n n
d dv du
uv u v
dx dx dx
d dw dv du
uvw uv uw vw
dx dx dx dx
v du dx u dv dx
d u
dx v v
d du
u nu
dx dx
−
= +
= + +
−
=
=
( )
( )
( )
( )
1
2
1 2
(Regla de la Cadena)
1
donde
dF dF du
dx du dx
du
dx dx du
dF du
dF
dx dx du
x f t
f t
dy dt
dy
dx dx dt f t y f t
= ⋅
=
=
=
′
= =
′ =
( )
( )
( )
( )
( )
( ) 1
1
ln
log
log
log
log 0, 1
ln
ln
a
a
u u
u u
v v v
du dx
d du
u
dx u u dx
d e du
u
dx u dx
e
d du
u a a
dx u dx
d du
e e
dx dx
d du
a a a
dx dx
d du dv
u vu u u
dx dx dx
−
= = ⋅
= ⋅
= ⋅ > ≠
= ⋅
= ⋅
= + ⋅ ⋅
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( )
( )
2
2
sen cos
cos sen
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
d du
u u
dx dx
=
= −
=
= −
=
= −
=
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
1
sen
1
1
cos
1
1
tg
1
1
ctg
1
si 1
1
sec
si 1
1
si 1
1
csc
si 1
1
1
vers
2
d du
u
dx dx
u
d du
u
dx dx
u
d du
u
dx dx
u
d du
u
dx dx
u
u
d du
u
u
dx dx
u u
u
d du
u
u
dx dx
u u
d
u
dx u u
∠ = ⋅
−
∠ = − ⋅
−
∠ = ⋅
+
∠ = − ⋅
+
+ >
∠ = ± ⋅
− < −
−
− >
∠ = ⋅
+ < −
−
∠ =
−
DERIV DE FUNCS TRIGO INVER
∓
2
du
dx
⋅
2
2
senh cosh
cosh senh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
=
=
=
= −
= −
= −
DERIVADA DE FUNCS HIPERBÓLICAS
1
2
-1
1
-1
2
1
2
1
2
1
1
2
1
senh
1
si cosh 0
1
cosh , 1
si cosh 0
1
1
tgh , 1
1
1
ctgh , 1
1
si sech 0, 0,
1
sech
1
d du
u
dx dx
u
u
d du
u u
dx dx u
u
d du
u u
dx dx
u
d du
u u
dx dx
u
u u
d du
u
dx dx
u u
−
−
−
−
−
−
= ⋅
+
+ >
±
= ⋅ >
− <
−
= ⋅ <
−
= ⋅ >
−
− > ∈
= ⋅
−
DERIVADA DE FUNCS HIP INV
∓
1
1
2
1
si sech 0, 0,1
1
csch , 0
1
u u
d du
u u
dx dx
u u
−
−
+ < ∈
= − ⋅ ≠
+
( )
( )
( ) ( )
( )
( )
2
2
0
1
ln
1
ln ln
1
ln ln ln 1
1
log ln ln 1
ln ln
log 2log 1
4
ln 2ln 1
4
u u
u
u
u
u
u u
a
a a
e du e
a
a
a du
a
a
a
ua du u
a a
ue du e u
udu u u u u u
u
udu u u u u
a a
u
u udu u
u
u udu u
=
>
=
≠
= ⋅ −
= −
= − = −
= − = −
= ⋅ −
= −
∫
∫
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS LOG & EXP
( ) ( )
{ } ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) [ ]
( ) ( )
( ) ( )
0
, , ,
,
b b b
a a a
b b
a a
b c b
a a c
b a
a b
a
a
b
a
b b
a a
f x g x dx f x dx g x dx
cf x dx c f x dx c
f x dx f x dx f x dx
f x dx f x dx
f x dx
m b a f x dx M b a
m f x M x a b m M
f x dx g x dx
f x g x x a
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
∫ ∫ ∫
∫ ∫
∫ ∫ ∫
∫ ∫
∫
∫
∫ ∫
INTEGRALES DEFINIDAS,
PROPIEDADES
[ ]
( ) ( ) si
b b
a a
b
f x dx f x dx a b
≤ <
∫ ∫
( ) ( )
( )
( )
1
Integración por partes
1
1
ln
n
n
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
u
u du n
n
du
u
u
+
=
=
± ± ± = ± ± ±
= −
= ≠ −
+
=
∫
∫ ∫
∫ ∫ ∫
∫ ∫
∫
∫
INTEGRALES
∫
2
2
sen cos
cos sen
sec tg
csc ctg
sec tg sec
csc ctg csc
udu u
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS TRIGO
tg ln cos ln sec
ctg ln sen
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫
∫
∫
∫
( )
2
2
2
2
1
sen sen 2
2 4
1
cos sen 2
2 4
tg tg
ctg ctg
u
udu u
u
udu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫
∫
∫
sen sen cos
cos cos sen
u udu u u u
u udu u u u
= −
= +
∫
∫
INTEGRALES DE FUNCS
( )
( )
2
2
2
2
2
2
sen sen 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
udu u u u
udu u u u
udu u u u
udu u u u
udu u u u u
u u u
udu u u u u
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − + −
= ∠ − ∠
∠ = ∠ + + −
∫
∫
∫
∫
∫
∫
TRIGO INV
csc cosh
u u u
= ∠ + ∠
INTEGRALES DE FUNCS HIP
2
2
senh cosh
cosh senh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu u
udu u
udu u
udu u
u udu u
u udu u
=
=
=
= −
= −
= −
∫
∫
∫
∫
∫
∫