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The document lists various trigonometric identities involving addition, subtraction, and multiplication of trigonometric functions like sine, cosine, and tangent. Some key identities include: (i) sin(x+y)=sinxcosy+cosxsiny, (ii) cos(x+y)=cosxcosy-sinxsiny, and (iii) tan(x+y)=tanxtany/(1-tanxtany).
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Identidades trigonometricas
This document is a guide on trigonometric identities from Professor Juan Carlos Morgado of Universidad de Valparaiso. It contains over 50 trigonometric identities to prove across two sections. The identities cover basic trig functions like sin, cos, tan as well as inverse trig functions like csc and cot. Students are asked to prove each identity through algebraic manipulation.
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This document lists various trigonometric identities including:
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This document is a guide on trigonometric identities from Professor Juan Carlos Morgado of Universidad de Valparaiso. It contains over 50 trigonometric identities to prove across two sections. The identities cover basic trig functions like sin, cos, tan as well as inverse trig functions like csc and cot. Students are asked to prove each identity through algebraic manipulation.
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1. integration-theory. Module-5 pdf
Integration is a reverse process of differentiation. The integral or primitive of a function f(x) with
respect to x is that function (x) whose derivative with respect to x is the given function f(x). It is
i. 0. dx = c
ii. 1.dx = x + c
iii. k.dx = kx + c (k R)
xn1
expressed symbolically as -
zf (x) dx (x)
iv. xn dx =
n 1
+ c (n –1)
v. z1 dx = log
x + c
Thus x e
vi. ex dx = ex + c
ax
The process of finding the integral of a function is called Integration and the given function is
vii. ax dx =
loge
a + c = ax loga e + c
called Integrand. Now, it is obvious that the operation of integration is inverse operation of differentiation. Hence integral of a function is also named as anti-derivative of that function.
Further we observe that-
viii. sin x dx = – cos x + c
ix. cos x dx = sin x + c
x. tan x dx = log sec x + c = – log cos x + c
d (x2 )
dx
2 x
xi. cot x dx = log sin x + c
d (x2 2) 2xV| 2xdx x2 constant
xii. sec x dx = log(secx + tanx) + c
dx = – log (sec x –tan x) + c
d 2
dx (x k) 2x|
= log tan
FGH xIJ+ c
So we always add a constant to the integral of function, which is called the constant of
xiii. cosec x dx = – log (cosec x + cot x) + c
Integration. It is generally denoted by c. Due to presence of this constant such an integral is called an Indefinite integral.
= log (cosec x – cot x) + c = log tan
xiv. sec x tan x dx = sec x + c
FGHxIJK+ c
If f(x), g(x) are two functions of a variable x and k is a constant, then-
(i) k f(x) dx = k f(x) dx.
(ii) [f(x) g(x)] dx = f(x)dx ± g(x) dx
(iii) d/dx ( f(x) dx) = f(x)
(iv) f(x)KJdx = f(x)
The following integrals are directly obtained from the derivatives of standard functions.
xv. cosec x cot x dx = – cosec x + c
xvi. sec2 x dx = tan x + c
xvii. cosec2 x dx = – cot x + c xviii. sinh x dx = cosh x + c
xix. cosh x dx = sinh x + c
xx. sech2 x dx = tanh x + c
xxi. cosech2 x dx = – coth x + c
xxii. sech x tanh x dx = – sech x + c
xxiii. cosech x coth x = – cosech x + c
1 1
FxI
eax
R 1FbI
xxiv. xxiv.
x2 + a2 dx =
a tan–1
GHa + c
= a2 b2
sin
STbx tan
GHaJK+ c
xxv. z 1
1
dx = log
FGx a + c
xxxv. zeax cos bx dx
x2 a2
2 a Hx aK
eax
xxvi. z 1
dx = 1 log FGa xIJ + c
= a2 b2
(a cos bx + b sin bx) + c
a2 x2
1
2 a Ha xK
FxI
= cos
STbx tan
1 b V+ c
xxvii. za2 x2 dx = sin–1
GHaJK+ c
FxI
Examples Integration of Function
xxviii. xxviii.
= – cos–1
1
dx = sinh–1
x2 a2
GHaJK+ c
FGxIJ+ c
Ex.1 Evaluate : zx–55 dx
Sol. x–55 dx
x54
= log (x +
) + c
= 54
+ c Ans.
xxix. z 1
dx = cosh–1
FGxIJ+ c
Ex.2 Evaluate :
zex2 1j2
x2 a2
= log (x +
HaK
) + c
Sol.
x
x4 2 x2 1
dx
x
xxx. xxx.
2 2 dx
= zx3 2x 1IJdx
za x
H xK
x4
= x +
2
a . sin–1
2
x + c
a
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This document provides formulas for integrals of common functions including polynomials, rational functions, radicals, logarithms, exponentials, and trigonometric functions. It includes basic integral formulas like the integral of x^n dx as well as more complex integrals involving combinations of functions. There are over 100 formulas presented in a table format organized by function type.
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respect to x is that function (x) whose derivative with respect to x is the given function f(x). It is
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ii. 1.dx = x + c
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zf (x) dx (x)
iv. xn dx =
n 1
+ c (n –1)
v. z1 dx = log
x + c
Thus x e
vi. ex dx = ex + c
ax
The process of finding the integral of a function is called Integration and the given function is
vii. ax dx =
loge
a + c = ax loga e + c
called Integrand. Now, it is obvious that the operation of integration is inverse operation of differentiation. Hence integral of a function is also named as anti-derivative of that function.
Further we observe that-
viii. sin x dx = – cos x + c
ix. cos x dx = sin x + c
x. tan x dx = log sec x + c = – log cos x + c
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dx
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xi. cot x dx = log sin x + c
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FGHxIJK+ c
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(ii) [f(x) g(x)] dx = f(x)dx ± g(x) dx
(iii) d/dx ( f(x) dx) = f(x)
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xv. cosec x cot x dx = – cosec x + c
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xx. sech2 x dx = tanh x + c
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FxI
eax
R 1FbI
xxiv. xxiv.
x2 + a2 dx =
a tan–1
GHa + c
= a2 b2
sin
STbx tan
GHaJK+ c
xxv. z 1
1
dx = log
FGx a + c
xxxv. zeax cos bx dx
x2 a2
2 a Hx aK
eax
xxvi. z 1
dx = 1 log FGa xIJ + c
= a2 b2
(a cos bx + b sin bx) + c
a2 x2
1
2 a Ha xK
FxI
= cos
STbx tan
1 b V+ c
xxvii. za2 x2 dx = sin–1
GHaJK+ c
FxI
Examples Integration of Function
xxviii. xxviii.
= – cos–1
1
dx = sinh–1
x2 a2
GHaJK+ c
FGxIJ+ c
Ex.1 Evaluate : zx–55 dx
Sol. x–55 dx
x54
= log (x +
) + c
= 54
+ c Ans.
xxix. z 1
dx = cosh–1
FGxIJ+ c
Ex.2 Evaluate :
zex2 1j2
x2 a2
= log (x +
HaK
) + c
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x
x4 2 x2 1
dx
x
xxx. xxx.
2 2 dx
= zx3 2x 1IJdx
za x
H xK
x4
= x +
2
a . sin–1
2
x + c
a
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This document contains mathematical formula tables covering a wide range of topics including:
- Greek alphabet
- Indices and logarithms
- Trigonometric, complex number, and hyperbolic identities
- Power series expansions
- Derivatives of common functions
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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This Dissertation explores the particular circumstances of Mirzapur, a region located in the
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Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
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of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
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Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
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For Upcoming Meetups Join Mysore Meetup Group - https://meetups.mulesoft.com/mysore/YouTube:- youtube.com/@mulesoftmysore
Mysore WhatsApp group:- https://chat.whatsapp.com/EhqtHtCC75vCAX7gaO842N
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- 1. Trigonometric Identities: ) sin (x + y) = sinx cosy + cosx siny (i) sin (x-y) = sinx cosy-cosx siny (ii) cos (x+ y) = cosx cosy sinx siny (iv) cos(x-y) =cosxcosy+sinx siny tanx+ tany tan(x + y)-tanx.tany (v) tanx-tany 1+ tanx.tany (vi) tan(x- y) (vii) cot(x+ +y)= - Cotx.coty-1 coty+cobx cot(x-y) = COtX.Coty+1 coty-cotx (vii)
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