Integrals 12th Notes - Free NCERT Class 12 Maths Chapter 7 Notes - Download PDF
Integrals belong to the 7 chapter of NCERT. The NCERT Class 12 Maths Chapter 7 notes entirely cover up the main portions of the chapter integrals. Class 12 Maths chapter 7 notes are predominantly focused on the important formulas and their needed derivations. A Class 12 Maths chapter 7 notes enable you to find the entire chapter in an easy way. Notes for Class 12 Maths chapter 7 is made by following the sequence of the chapter starting from the indefinite integrals to definite integrals. NCERT Notes for Class 12 Maths chapter 7 not only covers the NCERT notes but covers CBSE Class 12 Maths chapter 7 notes also.
After going through Class 12 integrals notes students can also refer to,
Also Read :
- NCERT Solutions for Class 12 Maths Chapter 7 Integrals
- NCERT Exemplar Class 12 Maths Chapter 7 Solutions Integrals
Adding or summing up the small parts to find the whole sum of the function, is the basic meaning of integral calculus.
Indefinite integral and definite integral are the two types of integrals that are found in integral calculus.
Thus the formula that gives this anti-derivative is called indefinite integral.
Integration as an Inverse Process of Differentiation
Integration is defined as the inverse of differentiation. Here we are given a function and asked to find it is primitive (the original function). Such a process is called anti-derivative or indefinite integral.
Let us see an example:-
We know that
Here the actual function is “sinx +c” using indefinite integral we found the actual equation from the derived one.
Thus we came to know that the integration of some functions is written as
Here f(x) is “integrand”, f(x)dx is “element of integration” , x is “variable of integration” and “c” is the integration constant.
Some of the formulas are listed below which will help us to solve the problems.
Geometrical Interpretation of Indefinite Integral
Let us assume that y=f(x) be a curve such that
Now if we give values like 4, 3, 2, 1, 0, -1, -2, -3 to arbitrary constant c then the equation will give
having loci as a parabola.
Thus the equation represents a family of curves.
Some properties of indefinite integral
Here we shall derive some properties of indefinite integrals.
(I) The process of integration and differentiation are inverses of each other in the sense of the following results:
where c is the arbitrary constant.
(II) Two indefinite integrals with the equal derivative lead to the same family of curves thus they are equivalent.
(V) Using property (III) and (IV) can be generalised to a finite number of functions f1, f2, f3, …, fn and the real numbers k1, k2, k3, …, kn gives
Find the anti-derivative (integral) of
Methods of Integration
We have many ways of solving integration. Mostly we solve by using
Integration by substitution
Integration by Partial Fraction
Integration by Parts
Integration by Substitution
The given integral can be transferred into another form by changing the independent variable x to t by substituting x = g (t).
Let us assume that
x=g(t) , so that dx/dt=g'(t)
Then we write dx=g’(t) dt
Example: Integrate w.r.t x, sin mx
We know that by differentiating mx we get m. Thus we can substitute mx=t so that mdx =dt
Integration by Trigonometric Identities
The identities are used to find the integral when integration involves some trigonometric functions.
Solution: We know that
Using this we get that
Integrals of Some Particular Function
Some of the formulas are listed below
Integration by Partial Fraction
The integration of rational function can be solved by using partial fraction.
Form of a rational function
formation of partial function
Integration by Parts
ILATE rule needs to be followed while doing integration by parts. According to ILATE rule, we can determine which function will be the 1st function and which will be the 2nd function.
I → Inverse function
L → Logarithmic function
A → Algebraic function
T → Trigonometric function
E → Exponential function
From the above rule, as an example, we can say that there are two functions like trigonometric function and logarithmic function. The logarithmic function will be treated as the first function and the trigonometric function will be treated as the second function.
The general expression follows as
“The integral of the product of two functions = (1st function) × (integral of the 2nd function) – Integral of [(differential coefficient of the 1st function) × (integral of the 2nd function)]”
Integration of special types function
The formulas are listed below
In indefinite, we saw that the result of the integration was not a unique value. But here indefinite integral we will find a unique value and it is well defined in a limited boundary. Generally, definite integral is denoted by
Her “a” is the lower limit and “b” is the upper limit of the integral. The value of the definite integral is computed by F(b) - F(a).
Basically in Definite integral consist of two cases as discussed below:
Definite Integral as a Limit of the Sum
If the definite integral
is the area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis. To evaluate this area, consider the region PRSQP between this curve, x-axis, and the ordinates x = a and x =b
From the figure, we find the general equation of definite integral as a limit of sum
Some Properties of Definite Integral
Let us see an example
This brings us to the end of the chapter.
Significance of NCERT Class 12 Maths Chapter 7 Notes
Class 12 integrals notes will be really helpful to revise the chapter and get a brief overview of the important topics. Also, Class 12 Math Chapter 7 Notes is useful for covering Class 12 CBSE syllabuses and also for competitive exams like BITSAT, JEE MAINS. Class 12 Math chapter 7 notes pdf download can be used for preparing in offline mode.
integrals Class 12 notes pdf download: link
NCERT Class 12 Notes Chapter Wise.
Subject Wise NCERT Exemplar Solutions
- NCERT Exemplar Class 12 Solutions
- NCERT Exemplar Class 12 Maths
- NCERT Exemplar Class 12 Physics
- NCERT Exemplar Class 12 Chemistry
- NCERT Exemplar Class 12 Biology
Subject Wise NCERT Solutions
- NCERT Solutions for Class 12 Mathematics
- NCERT Solutions for Class 12 Chemistry
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Biology
Frequently Asked Question (FAQs) - Integrals 12th Notes - Free NCERT Class 12 Maths Chapter 7 Notes - Download PDF
Question: From where we can download this notes>
The notes can be downloaded from integrals Class 12 notes pdf download.
Question: What textbook should be followed?
Class 12 math chapter 7 textbook should be followed by Class 12 Maths chapter 7 notes pdf download
Question: Write the different types of integrals in Maths.
There are two types of Integration. They are definite integral and indefinite integral.
Question: Define the fundamental theorem of calculus?
The fundamental theorem of calculus connects the concept of differentiation and integration of a function.
Question: What are the uses of integration?
In, NCERT Class 12 Maths chapter 7, integration is used to find the area, volumes, and central values of many things.
Question: Define integration?
The integration is the process of finding the anti-derivative function. Integration is defined as the reverse process of differentiation.
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