Integration in Maths – Definition, Formulas and Types, Class 12 Math Notes Study Material Download Free PDF

Integration Definition

The integration denotes the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured singularly. In a broad sense, in calculus, the idea of limit is used where algebra and geometry are implemented. Limits help us in the study of the result of points on a graph such as how they get closer to each other until their distance is almost zero. We know that there are two major types of calculus –

  • Differential Calculus
  • Integral Calculus

The concept of integration has developed to solve the following types of problems:

  • To find the problem function, when its derivatives are given.
  • To find the area bounded by the graph of a function under certain constraints.

These two problems lead to the development of the concept called the “Integral Calculus”, which consist of definite and indefinite integral. In calculus, the concept of differentiating a function and integrating a function is linked using the theorem called the Fundamental Theorem of Calculus. 

Maths Integration

In Maths, integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts. This method is used to find the summation under a vast scale. Calculation of small addition problems is an easy task which we can do manually or by using calculators as well. But for big addition problems, where the limits could reach to even infinity, integration methods are used. Integration and differentiation both are important parts of calculus. The concept level of these topics is very high. Hence, it is introduced to us at higher secondary classes and then in engineering or higher education. To get an in-depth knowledge of integrals, read the complete article here.

Integral Calculus

According to Mathematician Bernhard Riemann,

“Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.” Learn more about Integral calculus here.

Let us now try to understand what does that mean:

  • Take an example of a slope of a line in a graph to see what differential calculus is:

In general, we can find the slope by using the slope formula. But what if we are given to find an area of a curve? For a curve, the slope of the points varies, and it is then we need differential calculus to find the slope of a curve.

You must be familiar with finding out the derivative of a function using the rules of the derivative. Wasn’t it interesting? Now you are going to learn the other way round to find the original function using the rules in Integrating.

Integration – Inverse Process of Differentiation

We know that differentiation is the process of finding the derivative of the functions and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. So we can say that integration is the inverse process of differentiation or vice versa. The integration is also called the anti-differentiation. In this process, we are provided with the derivative of a function and asked to find out the function (i.e., primitive). 

We know that the differentiation of sin x is cos x. 

It is mathematically written as:

(d/dx) sinx = cos x …(1)

Here, cos x is the derivative of sin x. So, sin x is the antiderivative of the function cos x. Also, any real number “C” is considered as a constant function and the derivative of the constant function is zero. 

So, equation (1) can be written as

(d/dx) (sinx + C)= cos x +0

(d/dx) (sinx + C)= cos x 

Where “C” is the arbitrary constant or constant of integration.

Generally, we can write the function as follow:

(d/dx) [F(x)+C] = f(x), where x belongs to the interval I.

To represent the antiderivative of “f”, the integral symbol “∫” symbol is introduced. The antiderivative of the function is represented as ∫ f(x) dx. This can also be read as the indefinite integral of the function “f” with respect to x.

Therefore, the symbolic representation of the antiderivative of a function (Integration) is:

y = ∫ f(x) dx

∫ f(x) dx = F(x) + C.

Integrals in Maths

You have learned until now the concept of integration. You will come across, two types of integrals in maths:

Definite Integral

An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral.

A definite Integral is represented as:

\(\begin{array}{l}\int_{a}^{b} f(x)dx\end{array} \)

Indefinite Integral

Indefinite integrals are defined without upper and lower limits. It is represented as:

∫f(x)dx = F(x) + C

Where C is any constant and the function f(x) is called the integrand.

Integration Formulas

Check below the formulas of integral or integration, which are commonly used in higher-level maths calculations. Using these formulas, you can easily solve any problems related to integration.

Integration Formulas

Integration Examples

Solve some problems based on integration concept and formulas here.

Example 1: Find the integral of the function:

\(\begin{array}{l}\int_{0}^{3} x^{2}dx\end{array} \)

Solution:

\(\begin{array}{l}Given:\ \int_{0}^{3} x^{2}dx\end{array} \)

\(\begin{array}{l} = \left ( \frac{x^{3}}{3} \right )_{0}^{3}\end{array} \)

\(\begin{array}{l}= \left ( \frac{3^{3}}{3} \right ) – \left ( \frac{0^{3}}{3} \right )\end{array} \)

= 9

Example 2: Find the integral of the function: ∫x2 dx

Solution:

Given ∫x2 dx

= (x3/3) + C.

Example 3:

Integrate ∫ (x2-1)(4+3x)dx.

Solution:

Given: ∫ (x2-1)(4+3x)dx.

Multiply the terms, we get

∫ (x2-1)(4+3x)dx = ∫ 4x2+3x3-3x-4 dx

Now, integrate it, we get

∫ (x2-1)(4+3x)dx  = 4(x3/3) + 3(x4/4)- 3(x2/2) – 4x + C

The antiderivative of the given function ∫  (x2-1)(4+3x)dx is 4(x3/3) + 3(x4/4)- 3(x2/2) – 4x + C.

Frequently Asked Questions on Integration

Q1

What is integration?

The integration is the process of finding the antiderivative of a function. It is a similar way to add the slices to make it whole. The integration is the inverse process of differentiation.

Q2

What is the use of integration?

The integration is used to find the volume, area and the central values of many things.

Q3

What are the real-life applications of integration?

Integrations are much needed to calculate the centre of gravity, centre of mass, and helps to predict the position of the planets, and so on.

Q4

What is the fundamental theorem of calculus?

The fundamental theorem of calculus links the concept of differentiation and integration of a function.

Q5

Mention two different types of integrals in Maths.

Integration is one of the two main concepts of Maths, and the integral assigns a number to the function. The two different types of integrals are definite integral and indefinite integral.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Max Marks: 80

No.UnitsMarks
I.Relations and Functions08
II.Algebra10
III.Calculus35
IV.Vectors and Three – Dimensional Geometry14
V.Linear Programming05
VI.Probability08
Total Theory80
Internal Assessment20
Grand Total100

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals 

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Q1

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.

Q2

How many units are discussed in the CBSE Class 12 Maths Syllabus?

In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.

Q3

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.