Class 12 Maths Chapter 7 Integrals

If differentiation is about breaking things into tiny pieces, integration is about putting those pieces back together. This chapter will help you see how simple and logical integrals can be when understood step by step. You will learn how to find antiderivatives, how to compute definite integrals, and how to use different techniques to solve problems that initially seem difficult!

Whether you are preparing for boards, competitive exams, or your school tests, these notes will help you revise quickly and understand the concepts deeply.

Introduction

Integration is one of the two pillars of calculus. While differentiation deals with the rate of change of a function, integration performs the reverse task. It helps us accumulate quantities, find total values, compute areas, and reverse derivatives.

In simple words, integration is the process of finding the antiderivative. For any function f(x), integration finds a function F(x) such that the derivative of F(x) gives back f(x). Thus, F'(x) = f(x).

Class 12 mathematics focuses on two major types of integrals: indefinite integrals and definite integrals. You also learn techniques such as substitution, partial fractions, integration by parts and trigonometric simplifications.

Indefinite Integrals

An indefinite integral of a function f(x) is written as:

∫ f(x) dx = F(x) + C

Here:

• f(x) is the integrand
• dx indicates the variable of integration
• F(x) is the antiderivative
• C is the constant of integration

Example:
∫ 3x² dx = x³ + C

Basic Formulas of Integration

These standard integrals form the base for solving most questions:

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, where n ≠ -1

∫ 1 dx = x + C

∫ 1 ÷ x dx = ln|x| + C

∫ eˣ dx = eˣ + C

∫ aˣ dx = aˣ ÷ ln a + C

∫ sin x dx = -cos x + C

∫ cos x dx = sin x + C

∫ sec² x dx = tan x + C

∫ cosec² x dx = -cot x + C

∫ sec x tan x dx = sec x + C

∫ cosec x cot x dx = -cosec x + C

Methods of Integration

Because many functions are not directly integrable, different methods are used.

Integration by Substitution

This method works when a function becomes simpler by substituting t = g(x).

Formula:
∫ f(g(x)) g'(x) dx = ∫ f(t) dt

Example:
∫ 2x cos(x²) dx
Let t = x², dt = 2x dx
Result: ∫ cos t dt = sin t + C = sin(x²) + C

Integration by Parts

Used for products of two functions.

Formula:
∫ u v dx = u ∫ v dx − ∫ (du/dx) (∫ v dx) dx

u is chosen using the ILATE rule: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

Example:
∫ x eˣ dx
Take u = x, dv = eˣ dx
Result: x eˣ − eˣ + C

Partial Fractions

Used when the integrand is a rational function.

Example:
1 ÷ (x² - 1)
Break into: A ÷ (x - 1) + B ÷ (x + 1)

Trigonometric Integrals

Trigonometric identities help simplify integrals such as:

∫ sin² x dx, ∫ cos² x dx, ∫ sin x cos x dx

Example: sin² x = (1 - cos 2x) ÷ 2

Definite Integrals

A definite integral evaluates the integral between two limits a and b:

∫ from a to b f(x) dx = F(b) - F(a)

This value represents the net area between the curve y = f(x) and the x-axis from x = a to x = b.

Example:
∫ from 0 to 2 x dx = [x² ÷ 2] from 0 to 2 = 2

Geometrical Meaning

It represents the signed area under a curve:

• Above x-axis = positive area
• Below x-axis = negative area

To get the actual area, take the absolute value.

Properties of Definite Integrals

Some helpful properties:

∫ from a to a f(x) dx = 0

∫ from a to b f(x) dx = - ∫ from b to a f(x) dx

∫ from a to b f(x) dx + ∫ from b to c f(x) dx = ∫ from a to c f(x) dx

If f(x) is even:

∫ from -a to a f(x) dx = 2 ∫ from 0 to a f(x) dx

If f(x) is odd:

∫ from -a to a f(x) dx = 0

Applications of Integrals

Integrals are extremely useful in real life. Two major applications taught at Class 12 level are:

Area Under Curves

Formula:
Area = ∫ from a to b f(x) dx

Area Between Two Curves

If f(x) lies above g(x) in the interval [a, b]:

Area = ∫ from a to b (f(x) − g(x)) dx

Special Integrals

Some useful standard results:

∫ 1 ÷ √(a² - x²) dx = sin⁻¹ (x ÷ a) + C
∫ 1 ÷ (a² + x²) dx = (1 ÷ a) tan⁻¹ (x ÷ a) + C
∫ 1 ÷ (x² - a²) dx = (1 ÷ 2a) ln |(x - a) ÷ (x + a)| + C

These forms frequently appear in board and competitive exam questions.

You now have a complete and deeply explained set of notes on Integrals for Class 12. Each concept has been laid out step by step so that you can revise quickly and clearly. With consistent practice and repeated problem solving, integration becomes one of the most interesting and predictable chapters.

Keep these notes with you while revising, and you will feel more confident with every practice session. Believe in yourself and keep working steadily. You are absolutely capable of mastering this chapter.

FAQs

Q1. What is the main difference between indefinite and definite integrals?
Ans. An indefinite integral produces a general antiderivative with a constant of integration. A definite integral produces a specific numerical value representing the signed area under a curve between two limits.

Q2. Why is the constant C added in indefinite integrals?
Ans. Many functions have the same derivative because the derivative of a constant is zero. Therefore, while integrating, the constant of integration C is added to represent all possible antiderivatives.

Q3. How do we choose the method of integration?
Ans. Selection depends on the form of the function. Composite functions require substitution, products of functions use integration by parts, rational functions use partial fractions and trigonometric expressions require identities.

Q4. What is the meaning of a definite integral geometrically?
Ans. It represents the net signed area between the curve y = f(x) and the x-axis between two points x = a and x = b.

Q5. Are NCERT textbook questions enough for scoring well in integrals?
Ans. Yes, NCERT examples and exercises thoroughly cover all important concepts. Practicing additional questions from reference books strengthens speed and accuracy for board and competitive exams.