An integral assigns numbers to functions in mathematics to define displacement, area, volume, and other notions that arise by connecting infinitesimal data. The process of finding integrals is called integration. Definite integrals are used when the limits are defined to generate a unique value. Indefinite integrals are implemented when the boundaries of the integrand are not specified. In case, the lower limit and upper limit of the independent variable of a function are specified, its integration is described using definite integrals. Also, we have several integral formulas to deal with various definite integral problems in maths.

Table of Contents

Definite Integral Definition

The definite integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b] is expressed as

Here,

∫ = Integration symbol

a = Lower limit

b = Upper limit

f(x) = Integrand

dx = Integrating agent

Thus, ∫_a^b f(x) dx is read as the definite integral of f(x) with respect to dx from a to b.

Definite Integral as Limit of Sum

The definite integral of any function can be expressed either as the limit of a sum or if there exists an antiderivative F for the interval [a, b], then the definite integral of the function is the difference of the values at points a and b. Let us discuss definite integrals as a limit of a sum. Consider a continuous function f in x defined in the closed interval [a, b]. Assuming that f(x) > 0, the following graph depicts f in x.

The integral of f(x) is the area of the region bounded by the curve y = f(x). This area is represented by the region ABCD as shown in the above figure. This entire region lying between [a, b] is divided into n equal subintervals given by [x₀, x₁], [x₁, x₂], …… [x_r-1, x_r], [x_n-1, x_n].

Let us consider the width of each subinterval as h such that h → 0, x₀ = a, x₁ = a + h, x₂ = a + 2h,…..,x_r = a + rh, x_n = b = a + nh

and n = (b – a)/h

Also, n→∞ in the above representation.

Now, from the above figure, we write the areas of particular regions and intervals as:

Area of rectangle PQFR < area of the region PQSRP < area of rectangle PQSE ….(1)

Since. h→ 0, therefore x_r–  x_r-1→ 0. The following sums can be established as;

From the first inequality, considering any arbitrary subinterval [x_r-1, x_r] where r = 1, 2, 3….n, it can be said that s_n< area of the region ABCD n

Since, n→∞, the rectangular strips are very narrow, it can be assumed that the limiting values of s_n and S_n are equal, and the common limiting value gives us the area under the curve, i.e.,

From this, it can be said that this area is also the limiting value of an area lying between the rectangles below and above the curve. Therefore,

This is known as the definition of definite integral as the limit of sum.

Definite Integral Properties

Below is the list of some essential properties of definite integrals. These will help evaluate the definite integrals more efficiently.

• ∫_a^b f(x) dx = ∫_a^b f(t) d(t)
• ∫_a^b f(x) dx = – ∫_b^a f(x) dx
• ∫_a^a f(x) dx = 0
• ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx
• ∫_a^b f(x) dx = ∫_a^b f(a + b – x) dx
• ∫₀^a f(x) dx =  f(a – x) dx

Steps for calculating ∫_ab f(x) dx

Step 1: Find the indefinite integral ∫f(x) dx. Let this be F(x). There is no need to keep

integration constant C. This is because if we consider F(x) + C instead of F(x), we get

∫_a^b f(x) dx = [F(x) + C]_a^b = [F(b) + C] – [F(a) + C] = F(b) + C – F(a) – C = F(b) – F(a)

Thus, the arbitrary constant will not appear in evaluating the value of the definite integral.

Step 2: Calculate the value of F(b) – F(a) = [F(x)]_a^b

Hence, the value of ∫_a^b f(x) dx = F(b) – F(a)

Definite Integral by Parts

Below are the formulas to find the definite integral of a function by splitting it into parts.

• ∫₀^2a f (x) dx = ∫₀^a f (x) dx + ∫₀^a f (2a – x) dx
• ∫₀^2a f (x) dx = 2 ∫₀^a f (x) dx … if f(2a – x) = f (x).
• ∫₀^2a f (x) dx = 0 … if f (2a – x) = – f(x)
• ∫-a^a f(x) dx = 2 ∫₀^a f(x) dx … if f(- x) = f(x) or it is an even function
• ∫_-a^a f(x) dx = 0 … if f(- x) = – f(x) or it is an odd function

Definite Integral Examples

Example 1: 

Evaluate the value of ∫₂³ x² dx.

Solution:

Let I = ∫₂³ x² dx

Now, ∫x² dx = (x³)/3

Now, I =  ∫₂³ x² dx = [(x³)/3]₂³

= (3³)/3 – (2³)/3

= (27/3) – (8/3)

= (27 – 8)/3

= 19/3

Therefore, ∫₂³ x² dx = 19/3

Example 2:

Calculate: ∫₀^π/4 sin 2x dx

Solution:

Let I = ∫₀ ^π/4 sin 2x dx

Now, ∫ sin 2x dx = -(½) cos 2x

I = ∫₀ ^π/4 sin 2x dx

= [-(½) cos 2x]₀^π/4

= -(½) cos 2(π/4) – {-(½) cos 2(0)}

= -(½) cos π/2 + (½) cos 0

= -(½) (0) + (½)

= 1/2

Therefore, ∫₀ ^π/4 sin 2x dx = 1/2

Frequently Asked Questions on Definite Integral

Q1

What is a definite integral?

The definite integral has a unique value. A definite integral is denoted by ∫_a^b f(x) dx, where a is called the lower limit of the integral and b is called the upper limit of the integral.

Q2

What is the formula for definite integral?

The formula for calculating the definite integral of a function f(x) in the interval [a, b] is given by,
∫_a^b f(x) dx = F(b) – F(a)

Q3

What is a definite integral used for?

We can use definite integrals to find the area under, over, or between curves in calculus. If a function is strictly positive, the area between the curve of the function and the x-axis is equal to the definite integral of the function in the given interval. In the case of a negative function, the area will be -1 times the definite integral.

Q4

Can a definite integral be negative?

Yes, the value of a definite integral can be negative, positive or zero.

Q5

Do definite integrals have C?

No, definite integrals do not have C. As it is not required to add an arbitrary constant, i.e. C in case of definite integrals.

Neeraj Anand, Param Anand

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.

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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

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.