Chapter 7 Integrals Class 12 NCERT Solutions PDF Free

Are you looking for a Chapter 7 Integrals Class 12 NCERT Solutions PDF? So in today’s article, I will give you the solution for Class 12 Maths Chapter 7 Integration. The complete solution of all the chapters Ex 7.1, Ex 7.2, Ex 7.3, Ex 7.4 Ex 7.5, Ex 7.6, Ex 7.7, Ex 7.8, Ex 7.9, Ex 7.10 etc. will be found inside the same PDF.

Under this article, you will get the solution of all the exercises in one pdf and we will also know the topics of all the exercises. We will also understand very well all the topics that come under Integrals Class 12 NCERT Solutions PDF. Let us first know with the help of bullet points which exercises we will get in the integration.

• Exercise 7.1
• Exercise 7.2
• Exercise 7.3
• Exercise 7.4
• Exercise 7.5
• Exercise 7.6
• Exercise 7.7
• Exercise 7.8
• Exercise 7.9
• Exercise 7.10
• Miscellaneous Exercise

Here is a table summarizing some of the topics covered in Integrals for Class 12:

Let’s Get ‘Integrals Class 12 NCERT Solutions PDF’ For All Exercises

Integrals are an important part of the Class 12 Mathematics curriculum, and students are often required to solve complex problems involving integrals. To help students prepare for their exams, NCERT Solutions for Integrals Class 12 are available in PDF format.

Integrals Class 12 NCERT Solutions PDF provide students with step-by-step solutions to all the problems in the textbook. These solutions are designed to help students understand the concepts and solve problems with ease. The NCERT Solutions for Integrals Class 12 cover all the topics in the textbook, including indefinite integrals, definite integrals, integration by substitution, integration by parts, and more.

The PDF format of NCERT Solutions for Integrals Class 12 makes it easy for students to access the solutions on their smartphones, laptops, or tablets. They can get the PDF file and refer to it whenever they need help with a particular problem. The PDF format also allows students to take printouts of the solutions, making it easy to revise and practice problems offline.

Above you will be seeing Chapter 7 Integrals Class 12 NCERT Solutions PDF, with the help of which you can easily make notes of it. If you want, you can also save the PDF of this Integration Chapter 7.

Integrals Class 12 NCERT Solutions PDF is written by experts in the field of mathematics, ensuring that the solutions are accurate and reliable. These solutions are also regularly updated to keep up with the latest syllabus changes and exam patterns.

You Can Also Read: NCERT Maths Solutions for Class 12th

One of the key benefits of using NCERT Solutions for Integrals Class 12 is that they provide students with a comprehensive understanding of the subject. By referring to these solutions, students can learn how to approach different types of problems and develop a deeper understanding of the concepts.

In addition to NCERT Solutions for Integrals Class 12, students can also access other study materials such as sample papers, previous year’s question papers, and more. These resources can help students prepare for their exams and improve their performance in the subject.

NCERT Solutions for Integrals Class 12 in PDF format are an excellent resource for students preparing for their exams. They provide comprehensive solutions to all the problems in the textbook, helping students develop a deeper understanding of the subject. By using these solutions, students can improve their performance in the subject and achieve better results in their exams.

Miscellaneous Exercise Class 12 Integrals Chapter 7 PDF

The Miscellaneous Exercise for Class 12 Integrals is a set of problems that students can use to practice and improve their skills in solving indefinite and definite integrals. These exercises cover a wide range of topics and difficulty levels to help students prepare for their exams.

The Class 12 Integrals Miscellaneous Exercise PDF contains a collection of problems related to integration techniques such as integration by substitution, integration by parts, partial fractions, and trigonometric integrals. The PDF format allows students to access the exercises easily and practice them offline.

I have given this Chapter 7 Integrals Class 12 NCERT Solutions PDF which is of Miscellaneous Questions. All types of questions are present in various question papers, so it is very important to solve such pdf also. You can also save it on your mobile or you can make it online from this blog in your notes.

The exercises in the PDF file are designed to test the students’ understanding of the concepts and help them to apply their knowledge in solving practical problems. Each problem is accompanied by step-by-step solutions, which will help students to learn how to solve similar problems in the future.

The Miscellaneous Exercise PDF includes both subjective and objective questions. The subjective questions require students to show their work and provide detailed explanations, while the objective questions are designed to test their knowledge of the concepts and formulas.

The Miscellaneous Exercise Class 12 Integrals PDF is a valuable resource for students preparing for their Class 12 board exams. It helps them to practice a variety of problems and improve their understanding of the subject. Additionally, it provides them with an opportunity to test their skills and knowledge before the exams.

Indefinite Integrals

In Chapter 7 Integrals Class 12 NCERT Solutions PDF, indefinite integrals refer to finding the antiderivative of a function. An antiderivative is a function that, when differentiated, gives the original function. Indefinite integrals are also known as antiderivatives, and they play a crucial role in calculus.

To find the indefinite integral of a function, we use the integration symbol (∫) followed by the function we want to integrate, followed by the variable of integration. For example, the indefinite integral of f(x) with respect to x can be written as ∫f(x)dx.

The process of finding the indefinite integral involves reversing the process of differentiation. We use various integration techniques to find the antiderivative of a function. Some of the most common integration techniques are integration by substitution, integration by parts, and trigonometric substitution.

Indefinite integrals do not have limits of integration, unlike definite integrals. This means that the result of an indefinite integral is a family of functions, rather than a single value. The family of functions has a constant of integration, which can take any real value.

It is important to note that finding the antiderivative of a function is not always straightforward. In some cases, it may not be possible to find an antiderivative using the techniques we have at our disposal. In such cases, we can use numerical methods or approximation techniques to estimate the value of the integral.

Integration by Substitution

Integration by substitution is a technique used in calculus to simplify integrals. It involves substituting a variable in the integrand to transform the integral into a simpler form that can be easily evaluated.

The basic idea behind integration by substitution is to recognize a part of the integrand as a derivative of some other function. This can be achieved by looking for patterns in the integrand or by using trigonometric identities.

Once we have identified a part of the integrand that can be expressed as the derivative of another function, we substitute the variable with the new function. This helps us to simplify the integration and make it easier to evaluate.

The substitution variable is usually chosen so that the derivative of the new function is also present in the integrand. This allows us to simplify the integral further and eliminate the original variable.

The final step in integration by substitution is to convert the integral back into the original variable. This is done by replacing the substituted variable with its original expression.

Integration by substitution is a powerful technique that can be used to solve a wide range of integrals. It is particularly useful for integrals that involve nested functions or complicated expressions.

You will get all these in Integrals Class 12 NCERT Solutions PDF. There are also many questions on this which you will find in the PDF of Integrals Class 12 which I have given you above.

Integration by Parts

Integration by parts is a technique used in calculus to simplify integrals. It involves breaking down the integrand into two parts and applying a formula to transform the integral into a simpler form that can be easily evaluated.

The formula for integration by parts is:

∫u dv = uv – ∫v du

where u and v are functions of the variable of integration and ‘dv’ and du are their differentials.

To use integration by parts, we choose one part of the integrand as u and the other part as ‘dv’. We then apply the formula above to find the integral of the product of u and ‘dv’.

The choice of u and ‘dv’ is critical for the success of integration by parts. Generally, we choose u such that its derivative du is simpler than u itself. We then choose ‘dv’ to be the remaining part of the integrated.

After we have applied the formula and simplified the integral, we can repeat the process with the new integral until we obtain an integral that can be easily evaluated.

Integration by parts is a powerful technique that can be used to solve a wide range of integrals. It is particularly useful for integrals that involve products of functions or logarithmic functions.

Integration of Rational Functions

Integration of rational functions is an important topic in calculus that involves finding the antiderivative of a function that can be expressed as the ratio of two polynomials. A rational function can be written in the form:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials of x.

To integrate a rational function, we use the technique of partial fraction decomposition. This involves breaking down the rational function into simpler fractions that can be integrated separately.

The first step in partial fraction decomposition is to factor the denominator Q(x) into its irreducible factors. We then write the rational function as a sum of partial fractions, each with a numerator that is a constant or a polynomial of a degree less than the degree of the corresponding factor in the denominator.

For example, if Q(x) has distinct linear factors a and b, we can write:

f(x) = A / (x-a) + B / (x-b)

where A and B are constants that we need to find.

We can find the values of A and B by equating the coefficients of the terms on both sides of the equation. This gives us a system of linear equations that we can solve to find the values of A and B.

Once we have expressed the rational function as a sum of partial fractions, we can integrate each term separately. This involves using techniques such as substitution, integration by parts, and trigonometric substitution.

Integration of rational functions can be challenging, especially when the degree of the numerator is greater than or equal to the degree of the denominator. In such cases, we may need to use other techniques such as contour integration or numerical methods to evaluate the integral.

Integration of Trigonometric Functions

Integration of trigonometric functions is a fundamental topic in calculus that involves finding the antiderivative of functions that involve trigonometric functions such as sine, cosine, tangent, and their inverse functions.

The integrals of trigonometric functions can be evaluated using various techniques such as substitution, integration by parts, and trigonometric identities.

For example, to evaluate the integral of sin(x), we can use the substitution u = cos(x) and the identity sin^2(x) + cos^2(x) = 1 to obtain:

∫sin(x)dx = -cos(x) + C

where C is the constant of integration.

Similarly, to evaluate the integral of tan(x), we can use the substitution u = cos(x) and the identity 1 + tan^2(x) = sec^2(x) to obtain:

∫tan(x)dx = ln|sec(x)| + C

where C is the constant of integration.

The integrals of inverse trigonometric functions such as arcsin(x) and arctan(x) can also be evaluated using substitution and trigonometric identities.

Integration of trigonometric functions is an essential topic in calculus and is used in many areas of mathematics and physics. It is important for students to have a good understanding of the various techniques used to evaluate these integrals in order to succeed in calculus and related fields.

Integration of Exponential and Logarithmic Functions

Integration of exponential and logarithmic functions is an important topic in calculus that involves finding the antiderivative of functions that involve exponential functions such as e^x and logarithmic functions such as ln(x).

The integrals of exponential and logarithmic functions can be evaluated using various techniques such as substitution, integration by parts, and logarithmic differentiation.

For example, to evaluate the integral of e^x, we simply integrate it to obtain:

∫e^xdx = e^x + C

where C is the constant of integration.

Similarly, to evaluate the integral of ln(x), we can use the substitution u = ln(x) and integration by parts to obtain:

∫ln(x)dx = xln(x) – x + C

where C is the constant of integration.

The integrals of exponential and logarithmic functions are used in many areas of mathematics and science, including finance, physics, and engineering. They play a crucial role in modelling various phenomena such as population growth, radioactive decay, and electrical circuits.

Definite Integrals

Definite integrals are a fundamental topic in calculus that involve finding the area under a curve between two specific limits of integration. The definite integral of a function f(x) over an interval [a, b] is denoted by:

∫a^b f(x)dx

A definite integral is a number that represents the signed area between the graph of the function f(x) and the x-axis between limits a and b.

The definite integral can be evaluated using various techniques such as the Riemann sum, the midpoint rule, the trapezoidal rule, and Simpson’s rule. These techniques involve dividing the interval [a, b] into smaller subintervals and approximating the area under the curve by summing the areas of these smaller subintervals.

For example, the trapezoidal rule involves approximating the area under the curve by the sum of the areas of trapezoids that are formed by connecting the endpoints of the subintervals with straight line segments. The formula for the trapezoidal rule is:

∫a^b f(x)dx ≈ [(b-a)/2n][f(a) + 2f(a+(b-a)/n) + 2f(a+2(b-a)/n) + … + 2f(b-(b-a)/n) + f(b)]

where n is the number of subintervals.

The definite integral has many applications in mathematics, science, and engineering. It is used to calculate quantities such as the area under a probability distribution, the work done by a force, and the average value of a function over an interval.

definite integrals are a fundamental topic in calculus that involve finding the area under a curve between two specific limits of integration. The definite integral can be evaluated using various techniques such as the Riemann sum, the midpoint rule, the trapezoidal rule, and Simpson’s rule. These techniques are used to approximate the area under the curve by summing the areas of smaller subintervals. The definite integral has many applications in mathematics, science, and engineering and is an essential concept for students studying calculus.

Fundamental Theorem of Calculus

The fundamental theorem of calculus is a central concept in calculus that establishes the relationship between differentiation and integration. It has two parts, the first part being the derivative of the definite integral, and the second part being the evaluation of definite integrals using antiderivatives.

The first part of the theorem states that if f(x) is a continuous function on the interval [a, b], and F(x) is the antiderivative of f(x), then the derivative of the definite integral of f(x) over the interval [a, b] is given by:

d/dx ∫a^b f(x)dx = f(x)

In other words, the derivative of the definite integral of f(x) is equal to the original function f(x).

The second part of the theorem states that if f(x) is a continuous function on the interval [a, b], and F(x) is any antiderivative of f(x), then the definite integral of f(x) over the interval [a, b] is given by:

∫a^b f(x)dx = F(b) – F(a)

In other words, the definite integral of f(x) is equal to the difference between the values of the antiderivative F(x) at the upper and lower limits of integration.

The fundamental theorem of calculus has many applications in calculus and in various fields of science and engineering. It is used to evaluate definite integrals, find the area under a curve, calculate volumes and areas of revolution, and solve differential equations.

The fundamental theorem of calculus is a central concept in calculus that establishes the relationship between differentiation and integration. It has two parts, the first part being the derivative of the definite integral, and the second part being the evaluation of definite integrals using antiderivatives. The theorem has many applications in calculus and in various fields of science and engineering and is an essential concept for students studying calculus.

Properties of Definite Integrals

Definite integrals have several properties that make them useful in calculus and other areas of mathematics. Some of the important properties of definite integrals are:

1. Linearity: The definite integral is a linear operator, which means that it satisfies the following properties:∫a^b (f(x) + g(x))dx = ∫a^b f(x)dx + ∫a^b g(x)dx ∫a^b c.f(x)dx = c.∫a^b f(x)dx where f(x) and g(x) are functions, and c is a constant.
2. Change of limits: The limits of integration can be changed without changing the value of the integral. This property is known as the change of variables property and is expressed as ∫a^b f(x)dx = ∫c^d f(g(t)).g'(t)dtwhere g(t) is a differentiable function, and a=g(c) and b=g(d).
3. Symmetry: If the function f(x) is an even function, then the definite integral over a symmetric interval is equal to twice the integral over half the interval. If f(x) is an odd function, then the integral over a symmetric interval is zero.
4. Additivity: If an interval [a, b] is divided into two sub-intervals [a, c] and [c, b], then the definite integral over the interval [a, b] is equal to the sum of the definite integrals over the sub-intervals [a, c] and [c, b].
5. Monotonicity: If f(x) and g(x) are continuous functions on an interval [a, b], and f(x) ≤ g(x) for all x in [a, b], then ∫a^b f(x)dx ≤ ∫a^b g(x)dx.
6. Integration by parts: The definite integral of the product of two functions can be evaluated using integration by parts, which is expressed as ∫a^b u(x).v'(x)dx = [u(x).v(x)]|a^b – ∫a^b v(x).u'(x)dxwhere u(x) and v(x) are differentiable functions.

These properties of definite integrals are useful in evaluating integrals, finding areas under curves, calculating volumes and areas of revolution, and solving differential equations. They are also important for understanding the relationships between different functions and their integrals, and for developing a deeper understanding of calculus.

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Conclusion

The Chapter 7 Integrals Class 12 NCERT Solutions PDF provides a comprehensive guide to understanding the fundamental concepts of integration. The PDF covers various topics, including indefinite and definite integrals, integration by substitution, integration by parts, and trigonometric integrals, among others.

The PDF contains numerous examples and practice exercises that enable students to learn and master the concepts. It also provides step-by-step solutions for all the problems, making it easier for students to understand the processes involved in solving them.

By using this NCERT Solutions PDF, students can gain confidence in solving integration problems and improve their performance in the Class 12 board exams. It is an excellent resource for self-study and revision, and it helps students to understand the concepts in a simple and structured manner.

Overall we can say that Integrals Class 12 NCERT Solutions PDF is a valuable tool for students who are preparing for their exams or want to deepen their understanding of the subject. It is easily accessible and free, making it an excellent resource for students who want to excel in the field of math.

FAQ Related to Chapter 7 Integrals Class 12 NCERT Solutions PDF

FAQ (Frequently Asked Questions) related to Chapter 7 Integrals Class 12 NCERT Solutions PDF are a set of common questions and answers designed to help students understand the concepts covered in the PDF. These questions cover a range of topics related to integration, including the content of the PDF, the benefits of using it, and how to use it to prepare for exams. The FAQ is a helpful resource for students who want to deepen their understanding of integration or are looking for assistance in using the NCERT Solutions PDF effectively.