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This document contains 28 trigonometric identities that need to be demonstrated by Segundo Huaccha Alvarez. The identities involve trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant. They relate combinations of these functions through equalities, such as "sen x sec x = tg x". Segundo Huaccha Alvarez is asked to prove each of these 28 identities.
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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 =
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d (x2 )
dx
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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
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H xK
x4
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The document contains mathematical tables and formulas including:
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n 1
+ c (n –1)
v. z1 dx = log
x + c
Thus x e
vi. ex dx = ex + c
ax
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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.
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d (x2 )
dx
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xii. sec x dx = log(secx + tanx) + c
dx = – log (sec x –tan x) + c
d 2
dx (x k) 2x|
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= log (cosec x – cot x) + c = log tan
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FGHxIJK+ c
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(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)
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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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3. Additional trigonometric identities are presented for functions like sin(x ± y), cos(x ± y), and tan(x ± y) in terms of sums and differences of trigonometric functions.
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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.
Chapter 4
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Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
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.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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identidades ejercicios.docx
- 1. I.E. “SAN JUAN DE LLALLÁN” MATEMÁTICA 5° GRADO Segundo Huaccha Alvarez 1.- 2.- 3.- 4.- 5.-
- 2. I.E. “SAN JUAN DE LLALLÁN” MATEMÁTICA 5° GRADO Segundo Huaccha Alvarez Demostrar las siguientes identidades: 1. sen x ctg x = cos x 2. cos x tg x = sen x 3. ctg x sec x = cosec x 4. sen x sec x = tg x 5. cos x cosec x = ctg x 6. ctg x sec x sen x = 1 7. (1 - cos² x) cosec² x = 1 8. (1 - sen² x) sec² x = 1 9. ctg² x (1 - cos² x) = cos² x 10. (1 - cos² x) sec² x = tg² x 11. cosec x √1 - sen² x = ctg x 12. (1 + tg² x) cos² x = 1 13. (sec² x - 1) ctg² x = 1 14. (1 - cos² x)(1 + tg² x) = tg² x 15. cos x cosec x √sec² x - 1 = 1 16. sen² x (1 + ctg² x) = 1 17. (cosec² x - 1) tg² x = 1 18. (1 - cos² x)(1 + ctg² x) = 1 19. sen x sec x √cosec² x - 1 = 1 20. cos x √ctg² x + 1 = √cosec² x - 1 21. sen² x ctg² x + sen² x = 1 22. (1 + tg² x)(1 - sen² x) = 1 23. sen² x sec² x = sec² x - 1 24. cosec² x tg² x - 1 = tg² x 25. sec4 x - 1 = 2 tg² x + tg4 x 26. cosec4 x - 1 = 2 ctg² x + ctg4 x 27. (tg x cosec x)² - (sen x sec x)² = 1 28. (sec x ctg x)² - (cos x cosec x)² = 1
- 3. I.E. “SAN JUAN DE LLALLÁN” MATEMÁTICA 5° GRADO Segundo Huaccha Alvarez