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Integration of Trigonometric Functions - Formulas, Solved Examples

Integration of Trigonometric Functions - Formulas, Solved Examples

Edited By Komal Miglani | Updated on Jul 02, 2025 08:04 PM IST

The integration of trigonometric functions is one of the important parts of calculus, and it is applied to measure the change in the function at a certain point. Mathematically, it forms a powerful tool by which slopes of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These concepts of integration have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

Integration of Trigonometric Functions - Formulas, Solved Examples
Integration of Trigonometric Functions - Formulas, Solved Examples

In this article, we will cover the concept of Integration. This concept falls under the broader category of Calculus, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seventeen questions have been asked on this concept, including one in 2014, one in 2018, six in 2019, four in 2020, three in 2021, and two in 2023.

What is Integration?

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given. The rate of change of a quantity y concerning another quantity x is called the derivative or differential coefficient of y concerning x. Geometrically, the Differentiation of a function at a point represents the slope of the tangent to the graph of the function at that point.

Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.”

For example,

ddx(sinx)=cosxddx(x2)=2xddx(ex)=ex

In the above example, the function cos(x) is the derivative of sin(x). We say that sin(x) is an anti-derivative (or an integral) of cos(x). Similarly, x2 and ex are the antiderivatives (or integrals) of 2x and ex respectively.

Integration of Trigonometric Functions

Inverse trigonometric functions are known as arcus functions, cyclometric functions, or anti-trigonometric functions. These functions are used to get an angle for a given trigonometric value. It refers to the change in the value of the trigonometric function at a certain rate.

Inverse Trigonometric Functions

1. ddx(cosx)=sinxsinxdx=cosx+C
2. ddx(sinx)=cosxcosxdx=sinx+C
3. ddx(tanx)=sec2xsec2xdx=tanx+C
4. ddx(cotx)=csc2xcsc2xdx=cotx+C
5. ddx(secx)=secxtanxsecxtanxdx=secx+C
6. ddx(cscx)=cscxcotxcscxcotxdx=cscx+C

Integrals of tanx,cotx,secx,cosecx
7. ddx(log|sinx|)=cotxcotxdx=log|sinx|+C
8. ddx(log|cosx|)=tanxtanxdx=log|cosx|+C
9. ddx(log|secx+tanx|)=secxsecxdx=log|secx+tanx|+C
10. ddx(log|cscxcotx|)=cscxcscxdx=log|cscxcotx|+C

11. sec2xdx=tanx+Ccsc2xdx=cotx+C

Recommended Video Based on Integration of Trigonometric Functions


Solved Examples Based On Integration of Trigonometric Functions

Example 1: sin8xcos8x(12sin2xcos2x)dx is equal to I
1) 12sin2x+c
2) 12sin2x+c
3) 12sinx+c
4) sin2x+c

Solution

As learned in the concept

Integrals for Trigonometric functions -


ddx(cosx)=sinxsinxdx=cosx+csin8xcos8x(12sin2xcos2x)dx(sin4cos4x)(sin4x+cos4x)12sin2xcos2xdx(sin2+cos2x)(sin2cos2x)(sin4x+cos4x)12sin2xcos2xdx1(sin2cos2x)(sin4x+cos4x)12sin2xcos2xdx and (sin4x+cos4x)=(sin2x+cos2x)22(sin4xcos4x)=12(sin4xcos4x)1(sin2cos2x)(12sin2xcos2x)12sin2xcos2xdx(sin2cos2x)dxcos2xdx=12sin2x+C
Hence, the answer is the option 2.

Example 2: The integral 1+2cotx(cosecx+cotx)dx(0<x<π2) is equal to (where C is a constant of integration)
1)4log(sinx2)+C

2)2log(sinx2)+C

3)2log(cosX2)+C

4)4log(cosx2)+C

Solution

Integrals for Trigonometric functions -

Here we use trigonometric substitution to change this equation into a fundamental equation of integration.

ddx(cosx)=sinxsinxdx=cosx+c(1+2cotxcosecx+2cot2x)dx=(1+cot2x)+2cotxcosecx+cot2xdx=(cosec2x+2cotxcosecx+cot2x)dx=(cosecx+cotx)dx=ln|cscxcotx|+ln|sinx|+c=ln|1cosx|+c=ln|2sinx2|+c=ln|sin2x2|+ln2+c=2ln|sinx2|+c1


Hence, the answer is the option (2).

Example 3: Let αϵ(0,π/2) be fixed. If the integral tanx+tanαtanxtanαdx=A(x)cos2α+B(x)sin2α+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively :
1) x+α and loge|sin(x+α)|
2) xα and loge|sin(xα)|
3) xα and loge|cos(xα)|
4) x+α and loge|sin(xα)|

Solution
Integral of Trigonometric functions -

cotxdx=ln|sinx|+C

Here,


I=tanx+tanαtanxtanαdx=A(x)cos2α+B(x)sin2α+CI=sinxcosx+sinαcosαsinxcosxsinαcosαdx=sinxcosα+sinαcosxsinxcosαsinαcosxdx=sin(x+α)sin(xα)dx=sin(xα+2α)sin(xα)dx put xα=t;dx=dt=sin(t+2α)sin(t)dt=cos(2α)dt+sin(2α)cot(t)dt=(xα)cos2α+log|sin(xα)|sin2α+C

comparing with LHS

[A(x)=xαB(x)=loge|sin(xα)|]

Hence, the answer is the option 2.

Example 4: For x2nπ+1,nN(the set of natural numbers), the integral x2sin(x21)sin2(x21)2sin(x21)+sin2(x21)dx is equal to: (where c is a constant of integration) loge|12sec(x21)|+c
2) 12loge|sec2(x21)|+c
3) 12logc|sec(x212)|+c
4) logx|sec2(x212)|+c

Solution

Integral of Trigonometric functions -

tanxdx=ln|secx|+C

cotxdx=ln|sinx|+Csecxdx=ln|secx+tanx|+Ccosecxdx=ln|cosecxcotx|+C Put (x21)=t;2xdx=dtx2sin(x21)sin2(x21)2sin(x21)+sin2(x21)dx=122sin(t)sin2t2sin(t)+sin(2t)dtsin(2t)=2sin(t)cos(t)121cos(t)1+cos(t)dt12tan(t2)dtln|sec(t2)|+C replace t with (x21)ln|sec(x212)|+C

Hence, the answer is the option 4.

Example 5: If 0π2cotxcotx+cosecxdx=m(π+n), then mn is equal to :

1) -1

2) 1

3) -0.5

4) 0.5

Solution

Integration of trigonometric function -

sec2xdx=tanx+Ccsc2xdx=cotx+C


0π2cotxcotx+cscxdx=0π2cosxsinxcosxsinx+1sinxdx=0π2cosxcosx+1dx=0π22cos2x212cos2x2x+xdx=0π2(112sec2x2)dx=[xtanx2]0π2=12[π2]
So, m=12 and n=2

mn=12×2=1

Hence, the answer is the option (1).

Frequently Asked Questions (FAQs)

1. What is integration?

Integration is the reverse process of differentiation.

2. What is the other name of integration?

The other name of integration is antiderivative.

3. What is the integration of sin x?

 An integration of sin x is -cos x.

4. What is an integration of cos x?

An integration of cos x is sin x.

5. What is an integration of sec x tanx?

An integration of sec x tan x is sec x.

6. Why do we need special techniques for integrating trigonometric functions?
Trigonometric functions have unique properties and periodic behavior that make them challenging to integrate using standard methods. Special techniques, such as trigonometric identities and substitutions, are necessary to simplify these functions and find their antiderivatives accurately.
7. What is the fundamental principle behind integrating trigonometric functions?
The fundamental principle is to recognize patterns and use trigonometric identities to transform complex expressions into simpler forms that can be integrated more easily. This often involves rewriting products of trigonometric functions as sums or differences, or using substitution methods to simplify the integrand.
8. What is the key to integrating products of sines and cosines with different arguments, like sin(2x)cos(3x)?
The key is to use the product-to-sum formulas, which convert products of sines and cosines into sums of sines or cosines. This transformation often simplifies the integration process by breaking down complex products into more manageable terms.
9. Why does the integral of sec^2(x) equal tan(x), and how does this relate to derivatives?
The integral of sec^2(x) equals tan(x) because sec^2(x) is the derivative of tan(x). This relationship is a fundamental connection between integration and differentiation, and it's crucial for solving many trigonometric integrals.
10. What is the significance of the half-angle formulas in integrating even powers of sine and cosine?
Half-angle formulas are crucial for integrating even powers of sine and cosine because they allow us to reduce the power of the trigonometric function, often resulting in a more easily integrable expression. This technique is based on the algebraic properties of trigonometric functions and their periodic nature.
11. How does the integral of sin(x) differ from the derivative of sin(x)?
The integral of sin(x) is -cos(x) + C, while the derivative of sin(x) is cos(x). This difference highlights the inverse relationship between integration and differentiation, and the importance of the negative sign in the integral formula.
12. Why does integrating tan(x) result in -ln|cos(x)| instead of ln|sec(x)|?
Both -ln|cos(x)| and ln|sec(x)| are correct antiderivatives of tan(x). They differ by a constant, which is absorbed by the "+C" in indefinite integration. The choice between them is often based on preference or the specific context of the problem.
13. How does the method of integrating powers of sin(x) and cos(x) change when the exponents are odd versus even?
For odd powers, we typically use substitution methods (e.g., u = cos(x) for sin^n(x)). For even powers, we often use half-angle formulas to reduce the power and create integrable terms. This distinction arises from the different algebraic properties of odd and even functions.
14. What is the significance of the absolute value signs in the integral of sec(x)?
The absolute value signs in the integral of sec(x), which is ln|sec(x) + tan(x)| + C, ensure that the expression is defined for all values of x where sec(x) is defined. This accounts for the periodicity of the secant function and maintains the continuity of the antiderivative.
15. Why is the integral of sec(x)tan(x) simply sec(x), without any additional steps?
The integral of sec(x)tan(x) is sec(x) because sec(x)tan(x) is the derivative of sec(x). This is a direct application of the fundamental theorem of calculus and highlights the importance of recognizing derivative forms in integration.
16. How does the method of integration by parts apply to trigonometric functions?
Integration by parts is particularly useful for integrals involving products of trigonometric functions and polynomials or exponentials. It allows us to transform the integral into a form where one part becomes simpler while the other becomes more complex, often leading to a solvable recursive process.
17. Why is it sometimes necessary to use trigonometric substitutions in integration?
Trigonometric substitutions are necessary when dealing with certain algebraic expressions involving square roots. These substitutions transform complex algebraic expressions into trigonometric ones, which can often be integrated more easily using standard trigonometric integration techniques.
18. How do we handle integrals involving products of trigonometric and exponential functions?
Integrals involving products of trigonometric and exponential functions often require integration by parts. This technique allows us to leverage the derivatives of trigonometric functions and the integrals of exponential functions, often leading to a recursive process that can be solved systematically.
19. How do we approach integrals involving inverse trigonometric functions?
Integrals involving inverse trigonometric functions often require creative use of substitution methods or integration by parts. It's important to recognize the derivatives of inverse trigonometric functions and how they relate to algebraic expressions. Sometimes, rewriting the integrand in terms of the appropriate inverse function can simplify the problem.
20. How does the method of partial fractions apply to certain trigonometric integrals?
The method of partial fractions can be applied to trigonometric integrals that can be transformed into rational functions, often through substitutions like u = tan(x/2). This technique breaks down complex rational expressions into simpler terms that can be integrated individually, making it a powerful tool for solving certain types of trigonometric integrals.
21. Why is it important to consider the range of inverse trigonometric functions in integration?
Considering the range of inverse trigonometric functions is crucial in integration because these functions have restricted domains and ranges. This affects the interpretation of definite integrals and the validity of antiderivatives, especially when dealing with composite functions or expressions involving square roots.
22. Why is it sometimes necessary to use trigonometric substitutions for integrals involving square roots?
Trigonometric substitutions are necessary for integrals involving certain types of square roots because they can transform these algebraic expressions into trigonometric ones. This transformation often simplifies the integral, allowing us to use standard trigonometric integration techniques to solve problems that would be difficult or impossible to approach directly.
23. How does the integration of sin(x)cos(x) relate to the double angle formula?
The integration of sin(x)cos(x) is directly related to the double angle formula sin(2x) = 2sin(x)cos(x). By recognizing this relationship, we can simplify the integral to (1/2)sin^2(x) or -(1/4)cos(2x), demonstrating the power of trigonometric identities in integration.
24. How do we approach integrals involving secant and cosecant functions?
Integrals involving secant and cosecant often require special techniques. Common approaches include using trigonometric identities to rewrite the expression in terms of sine and cosine, or recognizing patterns that lead to known antiderivatives. The key is to transform these functions into more familiar or integrable forms.
25. How does the concept of odd and even functions impact the integration of trigonometric functions?
The odd or even nature of trigonometric functions affects their behavior over different intervals and can simplify integration. For example, the integral of an odd function over a symmetric interval centered at zero is always zero, while even functions may have special properties that simplify their integrals over certain intervals.
26. What is the relationship between the integrals of sin^n(x) and cos^n(x)?
The integrals of sin^n(x) and cos^n(x) are closely related due to the symmetry of these functions. They often follow similar patterns in their integration techniques, with the main difference being the starting point of the recursion (cos(x) for sin^n(x) and sin(x) for cos^n(x)).
27. How do we handle integrals involving products of trigonometric functions with different periods?
Integrals involving products of trigonometric functions with different periods often require the use of product-to-sum formulas or other trigonometric identities. These techniques help to break down the complex product into simpler terms that can be integrated individually.
28. Why is it important to consider the domain when integrating trigonometric functions?
Considering the domain is crucial because trigonometric functions have periodic behavior and may have restricted domains for certain operations. This affects the validity of antiderivatives and the interpretation of definite integrals, especially when dealing with inverse trigonometric functions or expressions involving square roots.
29. How does the integration of tan^2(x) relate to the Pythagorean identity?
The integration of tan^2(x) is closely related to the Pythagorean identity tan^2(x) + 1 = sec^2(x). By using this identity, we can rewrite tan^2(x) as sec^2(x) - 1, which simplifies the integration process and connects it to the known integral of sec^2(x).
30. What is the significance of the reduction formulas in trigonometric integration?
Reduction formulas are crucial for integrating higher powers of trigonometric functions. They provide a systematic way to reduce the power of the integrand, eventually leading to a known integral. These formulas demonstrate the recursive nature of many trigonometric integrals and offer a structured approach to solving complex problems.
31. How do we approach integrals involving products of sine and cosine with different arguments?
For integrals involving products of sine and cosine with different arguments, we typically use product-to-sum formulas. These formulas transform the product into a sum of sines or cosines, which are often easier to integrate. This technique is based on fundamental trigonometric identities and is essential for simplifying complex trigonometric integrals.
32. Why is the substitution u = tan(x/2) useful in certain trigonometric integrals?
The substitution u = tan(x/2), known as the Weierstrass substitution, is useful because it can transform trigonometric functions of x into rational functions of u. This conversion often simplifies complex trigonometric integrals into more manageable algebraic forms, making them easier to solve using standard integration techniques.
33. How does the integration of sec(x) relate to the natural logarithm function?
The integration of sec(x) results in ln|sec(x) + tan(x)| + C, which involves the natural logarithm function. This relationship arises from the unique properties of the secant function and demonstrates how transcendental functions like logarithms can emerge from trigonometric integration.
34. What is the significance of the integral of 1/(a^2 + x^2) in trigonometric integration?
The integral of 1/(a^2 + x^2) is significant because it leads to the arctangent function, (1/a)arctan(x/a). This integral form appears frequently in trigonometric substitutions and is crucial for solving many problems involving inverse trigonometric functions.
35. Why is it important to recognize patterns in trigonometric integrals?
Recognizing patterns in trigonometric integrals is crucial because many complex integrals can be simplified or solved by relating them to known, simpler forms. Pattern recognition helps in applying appropriate techniques, such as substitutions or identities, and often reveals connections between seemingly different problems.
36. How does the concept of periodicity affect the definite integration of trigonometric functions?
The periodicity of trigonometric functions significantly affects definite integration. When integrating over full periods or symmetric intervals, many trigonometric integrals simplify or vanish due to the cyclic nature of these functions. Understanding periodicity helps in evaluating definite integrals more efficiently and interpreting their results correctly.
37. What is the role of symmetry in simplifying trigonometric integrals?
Symmetry plays a crucial role in simplifying trigonometric integrals. Odd functions integrated over symmetric intervals centered at zero yield zero, while even functions often have special properties that can be exploited. Recognizing and utilizing symmetry can significantly reduce the complexity of many trigonometric integration problems.
38. Why is it sometimes necessary to use complex exponentials in trigonometric integration?
Complex exponentials can be useful in trigonometric integration because they provide an alternative representation of trigonometric functions through Euler's formula (e^(ix) = cos(x) + isin(x)). This representation can sometimes simplify complex trigonometric expressions and make them more amenable to integration techniques.
39. How does the integration of sin(x)cos(x) relate to the power reduction formulas?
The integration of sin(x)cos(x) is directly related to the power reduction formula sin(2x) = 2sin(x)cos(x). This relationship demonstrates how power reduction formulas can simplify products of trigonometric functions into more easily integrable forms, often involving functions of double angles.
40. What is the significance of the integral of tan(x) in relation to logarithmic functions?
The integral of tan(x) is significant because it results in -ln|cos(x)| + C, which involves a logarithmic function. This relationship highlights the connection between trigonometric and logarithmic functions and demonstrates how transcendental functions can arise from seemingly simple trigonometric integrals.
41. How do we handle integrals involving products of trigonometric functions and polynomials?
Integrals involving products of trigonometric functions and polynomials often require a combination of techniques. For lower degree polynomials, integration by parts is commonly used. For higher degrees, it may be necessary to use trigonometric substitutions or reduce the problem using trigonometric identities before integration.
42. Why is it important to understand the relationship between trigonometric functions and their reciprocals in integration?
Understanding the relationship between trigonometric functions and their reciprocals (e.g., sin(x) and csc(x), cos(x) and sec(x)) is crucial in integration because it allows for the transformation of complex integrals into simpler forms. This relationship often helps in recognizing patterns and applying appropriate integration techniques.
43. What is the significance of the double angle formulas in trigonometric integration?
Double angle formulas are significant in trigonometric integration because they allow us to transform products of trigonometric functions into sums or differences of functions with simpler arguments. This transformation often simplifies the integration process and is particularly useful for integrals involving products of sines and cosines.
44. How do we approach integrals involving powers of secant and tangent functions?
Integrals involving powers of secant and tangent functions often require a combination of techniques. For odd powers of secant, we typically use substitution methods. For even powers, reduction formulas or trigonometric identities are often employed. The key is to recognize patterns and transform the integrand into more manageable forms.
45. How does the concept of odd and even functions impact the integration of products of trigonometric functions?
The odd or even nature of trigonometric functions affects the integration of their products. When integrating the product of an odd and an even function over a symmetric interval, the result is often zero. Understanding these properties can simplify many integrals and provide insights into the behavior of trigonometric functions over different intervals.
46. What is the role of the half-angle formulas in simplifying certain trigonometric integrals?
Half-angle formulas play a crucial role in simplifying certain trigonometric integrals, particularly those involving even powers of sine or cosine. These formulas allow us to reduce the power of the trigonometric function, often resulting in a more easily integrable expression. They are essential tools for tackling complex trigonometric integrals.
47. How do we handle integrals involving products of trigonometric functions with different frequencies?
Integrals involving products of trigonometric functions with different frequencies often require the use of product-to-sum formulas or other trigonometric identities. These techniques help to break down the complex product into simpler terms that can be integrated individually, making use of the periodic nature of trigonometric functions.
48. How does the integration of cot(x) relate to the natural logarithm function?
The integration of cot(x) results in ln|sin(x)| + C, which involves the natural logarithm function. This relationship demonstrates how logarithmic functions can emerge from trigonometric integration and highlights the interconnectedness of different classes of functions in calculus.
49. What is the significance of the reduction formulas for sin^n(x) and cos^n(x)?
The reduction formulas for sin^n(x) and cos^n(x) are significant because they provide a systematic way to integrate higher powers of these functions. These formulas work by reducing the power of the trigonometric function in each step, eventually leading to a known integral. They demonstrate the recursive nature of many trigonometric integrals.
50. How do we approach integrals involving products of trigonometric functions and rational functions?
Integrals involving products of trigonometric functions and rational functions often require a combination of techniques. We might use trigonometric substitutions to convert the rational part into a trigonometric expression, or use partial fraction decomposition after a suitable substitution. The key is

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Arrange the following Cobalt complexes in the order of incresing Crystal Field Stabilization Energy (CFSE) value. Complexes :  

\mathrm{\underset{\textbf{A}}{\left [ CoF_{6} \right ]^{3-}},\underset{\textbf{B}}{\left [ Co\left ( H_{2}O \right )_{6} \right ]^{2+}},\underset{\textbf{C}}{\left [ Co\left ( NH_{3} \right )_{6} \right ]^{3+}}\: and\: \ \underset{\textbf{D}}{\left [ Co\left ( en \right )_{3} \right ]^{3+}}}

Choose the correct option :
Option: 1 \mathrm{B< C< D< A}
Option: 2 \mathrm{B< A< C< D}
Option: 3 \mathrm{A< B< C< D}
Option: 4 \mathrm{C< D< B< A}

The type of hybridisation and magnetic property of the complex \left[\mathrm{MnCl}_{6}\right]^{3-}, respectively, are :
Option: 1 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 2 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 3 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 4 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 5 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 6 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 7 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 8 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 9 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 10 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 11 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 12 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 13 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 14 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 15 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 16 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
The number of geometrical isomers found in the metal complexes \mathrm{\left[ PtCl _{2}\left( NH _{3}\right)_{2}\right],\left[ Ni ( CO )_{4}\right], \left[ Ru \left( H _{2} O \right)_{3} Cl _{3}\right] \text { and }\left[ CoCl _{2}\left( NH _{3}\right)_{4}\right]^{+}} respectively, are :
Option: 1 1,1,1,1
Option: 2 1,1,1,1
Option: 3 1,1,1,1
Option: 4 1,1,1,1
Option: 5 2,1,2,2
Option: 6 2,1,2,2
Option: 7 2,1,2,2
Option: 8 2,1,2,2
Option: 9 2,0,2,2
Option: 10 2,0,2,2
Option: 11 2,0,2,2
Option: 12 2,0,2,2
Option: 13 2,1,2,1
Option: 14 2,1,2,1
Option: 15 2,1,2,1
Option: 16 2,1,2,1
Spin only magnetic moment of an octahedral complex of \mathrm{Fe}^{2+} in the presence of a strong field ligand in BM is :
Option: 1 4.89
Option: 2 4.89
Option: 3 4.89
Option: 4 4.89
Option: 5 2.82
Option: 6 2.82
Option: 7 2.82
Option: 8 2.82
Option: 9 0
Option: 10 0
Option: 11 0
Option: 12 0
Option: 13 3.46
Option: 14 3.46
Option: 15 3.46
Option: 16 3.46

3 moles of metal complex with formula \mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{3} gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of CO in the complex is_______.
(Round off to the nearest integer)
 

The overall stability constant of the complex ion \mathrm{\left [ Cu\left ( NH_{3} \right )_{4} \right ]^{2+}} is 2.1\times 10^{1 3}. The overall dissociation constant is y\times 10^{-14}. Then y is ___________(Nearest integer)
 

Identify the correct order of solubility in aqueous medium:

Option: 1

Na2S > ZnS > CuS


Option: 2

CuS > ZnS > Na2S


Option: 3

ZnS > Na2S > CuS


Option: 4

Na2S > CuS > ZnS


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