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Class 12 Chapter 7

Class 12 Maths Chapter 7: Integrals

Integration is one of the fundamental concepts of Calculus and serves as the inverse process of differentiation. The Integrals chapter introduces techniques for evaluating indefinite and definite integrals and explains their applications in finding areas under curves, accumulated quantities, and solving mathematical problems. It forms the foundation for advanced topics such as Applications of Integrals and Differential Equations, making it an essential chapter for CBSE Board, JEE Main, and JEE Advanced.

At ALLEN, Class 12 Maths Chapter 7: Integrals study materials are designed to simplify Integrals through concise theory, important formulas, standard integration techniques, solved examples, and exam-oriented tips. Whether you're revising before board exams or preparing for competitive entrance examinations, these notes provide a structured approach for mastering the chapter efficiently.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	4–5 Hours
Key Concepts Covered	Integration by Substitution, Integration by Parts, Partial Fractions, Indefinite Integrals, Definite Integrals

2.0Chapter 7 Smart Study Material by ALLEN Experts: Download PDFsResource

Status
NCERT Solutions	Available Soon
Important Questions	Available Soon
Formula Sheet PDF	Available Soon
Previous Years' Questions (PYQs)	Available Soon
Mock Test	Available Soon
Practice Worksheets	Available Soon
Quick Revision Notes PDF	Available Soon
Mind Maps	Available Soon

Integral Calculus

Integral Calculus is the branch of calculus focused on accumulation, such as finding areas under curves, volumes, and total changes. It involves indefinite integrals (antiderivatives) and definite integrals (numerical values over intervals). Integral calculus essentially reverses differentiation, helping us reconstruct a function from its rate of change. Widely used in physics, engineering, and economics, it allows us to analyze continuous change and solve real-world problems involving motion, growth, and distribution.

1.0What is Integral Calculus?

Integral Calculus is the branch of calculus concerned with accumulation—how small pieces add up to make a whole. If differential calculus is about breaking things down (derivatives), then integral calculus is about putting things back together (integrals).

It primarily deals with:

Indefinite Integrals: Represent antiderivatives of functions.
Definite Integrals: Represent actual values, like area or total change.

Related Video:

2.0Indefinite Integrals

An indefinite integral is the reverse process of differentiation. It gives a general form of all antiderivatives of a function and includes an arbitrary constant of integration C, since derivatives of constants are zero.

$\int f (x) d x = F (x) + C$

Here:

f(x) is the integrand,
F(x) is any function such that F'(x) = f(x),
C is the constant of integration.

Example:

$\int 2 x d x = x^{2} + C$

Indefinite integrals are functions, not numbers. They're used when you're looking for a general formula rather than a specific value.

3.0Definite Integrals

A definite integral gives the net accumulation of a quantity over an interval [a, b]. It produces a numerical value and has direct geometric meaning—like the area under a curve.

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

Where:

F(x) is the antiderivative of f(x),
a and b are the limits of integration.

Key Properties:

$\int_{a}^{a} f (x) d x = 0$

$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$

If f(x) $\geq$ 0, the integral represents area.

Example:

$\int_{0}^{2} x^{2} d x = [\frac{x ^{3}}{3}]_{0}^{2} = \frac{8}{3}$

Definite integrals are numbers, often used to compute areas, volumes, and total change.

1.0The Core Idea: Summing Infinitesimally Small Quantities

To find the area under a curve y = f(x), we slice it into thin rectangles and add their areas. As we take more slices, we get closer to the true area. This infinite sum leads us to a definite integral:

$\int_{a}^{b} f (x) d x$

Where:

f(x): the function
a, b: the limits of integration
dx: an infinitesimally small width

2.0The Fundamental Theorem of Calculus

This theorem connects differentiation and integration:

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

Here, F(x) is an antiderivative of f(x), meaning F'(x) = f(x).

3.0Common Integration Techniques

1. Substitution (u-substitution):
Use when an integral looks like a chain rule in reverse.

$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$

2. Integration by Parts:

$\int u d v = uv - \int v d u$

3. Partial Fractions:
Break complex rational expressions into simpler terms.

4. Trigonometric Identities:
Useful for integrating sin²x, cos²x, etc.

5. Special Integrals:
Learn standard forms like:

$\int \frac{1}{1 + x ^{2}} d x = tan^{- 1} x + C$

4.0Applications of Integral Calculus

Area under curves
Volume of solids of revolution
Displacement and distance from velocity
Center of mass and moment of inertia
Probability distributions (in stats)

5.0Solved Examples on Integral Calculus

Example 1: Evaluate: $\int \frac{x}{1 + x ^{2}} d x$

Solution:
Let u = $1 + x^{2} \Rightarrow d u = 2 x d x$

So,

$\frac{d u}{2} = x d x$

$\int \frac{x}{1 + x ^{2}} d x = \int \frac{1}{u} \cdot \frac{d u}{2} = \frac{1}{2} \int u^{- 1/2} d u$

$= \frac{1}{2} \cdot \frac{u ^{1/2}}{1/2} = u = 1 + x^{2} + C$

Example 2: Evaluate: $\int_{0}^{π} \frac{x s i n x}{1 + c o s ^{2} x} d x$

Solution:

Use the property:

$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$

Let:

$I = \int_{0}^{π} \frac{x s i n x}{1 + c o s ^{2} x} d x$

Apply the property:

$I = \int_{0}^{π} \frac{( π - x ) s i n x}{1 + c o s ^{2} x} d x$

Add both:

$2 I = \int_{0}^{π} \frac{x s i n x + ( π - x ) s i n x}{1 + c o s ^{2} x} d x = \int_{0}^{π} \frac{π s i n x}{1 + c o s ^{2} x} d x$

$2 I = π \int_{0}^{π} \frac{s i n x}{1 + c o s ^{2} x} d x$

Let $u = cos x \Rightarrow d u = - sin x d x$

Change limits:
When x = 0 $\Rightarrow u = 1,$
When $x = π \Rightarrow u = - 1$

So:

$2 I = - π \int_{1}^{- 1} \frac{1}{1 + u ^{2}} d u = π \int_{- 1}^{1} \frac{1}{1 + u ^{2}} d u$

This is a standard integral:

$π [tan^{- 1} u]_{- 1}^{1} = π (tan^{- 1} 1 - tan^{- 1} (- 1)) = π (\frac{π}{4} + \frac{π}{4}) = π \cdot \frac{π}{2} \Rightarrow 2 I = \frac{π ^{2}}{2} \Rightarrow I = \frac{π ^{2}}{4}$

Example 3: Evaluate: $\int_{0}^{3} [x] d x (where [x] is the greatest integer function)$

Solution:

Break it at integers:

$\int_{0}^{3} [x] d x = \int_{0}^{1} 0 d x + \int_{1}^{2} 1 d x + \int_{2}^{3} 2 d x$ = 0 + 1 + 2 = 3

Example 4: Evaluate:

$\int x ln x d x$

Solution:

Use integration by parts:
Let u = $ln x, d v = x d x$
Then $d u = \frac{1}{x} d x, v = \frac{x ^{2}}{2}$

$\int x ln x d x = \frac{x ^{2}}{2} ln x - \int \frac{x ^{2}}{2} \cdot \frac{1}{x} d x = \frac{x ^{2}}{2} ln x - \int \frac{x}{2} d x = \frac{x ^{2}}{2} ln x - \frac{x ^{2}}{4} + C$

6.0Practice Questions

1. Find the integral: $\int (3 x^{2} + 2 x + 1) d x$

2. Evaluate: $\int_{0}^{2} x^{2} d x$

3. Find the area under the curve $y = sin x from x = 0 to x = π .$

4. Solve using substitution: $\int x x^{2} + 1 d x$

5. Evaluate using integration by parts: $\int x e^{x} d x$

Also Read:

Integrals of Particular Functions	Application of Integrals	Integration by Substitution
Integrating Factor	Differentiation and Integration	Integration by Parts
important integration formulas for jee	Integration by Partial Fractions	definite and indefinite integration

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of indefinite and definite integrals.
Apply standard integration formulas and properties.
Solve integrals using substitution, partial fractions, and integration by parts.
Evaluate definite integrals using their properties.
Simplify complex integration problems using appropriate techniques.
Build a strong foundation for Applications of Integrals and Differential Equations.
Solve JEE Main, JEE Advanced, and CBSE Board questions with confidence.

2.05-Minute Quick Revision

Must-Revise Concepts

Indefinite Integrals
Definite Integrals
Fundamental Theorem of Calculus
Standard Integration Formulas
Integration by Substitution
Integration by Parts
Partial Fractions
Properties of Definite Integrals
Trigonometric Integrals
Rational and Irrational Functions

Important Formula Recall

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
$\int e^{x} d x = e^{x} + C$
$\int sin x d x = - cos x + C$
$\int cos x d x = sin x + C$
$\int_{a}^{b} f (x), d x = - \int_{b}^{a} f (x), d x$
$\int_{a}^{a} f (x), d x = 0$

Important Properties / Shortcut Tips

Always identify the appropriate integration technique before solving.
Use substitution when the integrand contains a composite function.
Apply integration by parts for products of functions.
Learn symmetry properties of definite integrals to reduce calculations.
Remember to include the constant of integration in indefinite integrals.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Standard Integration Formulas	High	Easy–Moderate
Integration by Substitution	Very High	Moderate
Integration by Parts	Very High	Moderate
Partial Fractions	High	Moderate
Definite Integrals & Properties	Very High	Moderate–High
Trigonometric Integrals	High	Moderate
Mixed Integration Problems	Very High	High

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Forgetting the constant of integration	Add + C in every indefinite integral.
Choosing the wrong integration method	Identify whether substitution, parts, or partial fractions is suitable.
Incorrect limits in definite integrals	Substitute the upper and lower limits carefully after integration.
Sign errors in trigonometric integrals	Memorize standard integral formulas thoroughly.
Ignoring definite integral properties	Use symmetry and interval properties to simplify lengthy calculations.
Algebraic simplification mistakes	Simplify the integrand before applying integration techniques.
Confusing differentiation and integration formulas	Revise both concepts together for better accuracy.

6.0ALLEN Faculty Tips

Memorize standard integration formulas before attempting advanced problems.
Practice identifying the most suitable integration technique.
Revise all important properties of definite integrals regularly.
Focus on substitution and integration by parts, as they are frequently tested.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.
Avoid lengthy calculations by applying symmetry properties whenever possible.

7.0PYQ Trend Analysis

Integrals is one of the highest-weightage chapters in Calculus and is frequently tested in JEE Main, JEE Advanced, and CBSE Board examinations through concept-based and application-oriented questions.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Standard Integration	1–2	Easy–Moderate
Integration by Substitution	1–2	Moderate
Integration by Parts	1–2	Moderate
Partial Fractions	1	Moderate
Definite Integrals	2–3	Moderate–High
Properties of Definite Integrals	1–2	Moderate

8.0Most Common Question Types

Evaluating indefinite integrals using standard formulas.
Solving integrals using substitution and integration by parts.
Evaluating definite integrals using properties.
Integrating rational and trigonometric functions.
Simplifying complex integrals using algebraic manipulation.
Applying partial fraction decomposition.
Solving mixed integration problems.
Concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise all standard integration formulas.
Practice substitution-based integrals.
Solve integration by parts questions.
Revise partial fraction decomposition techniques.
Learn important properties of definite integrals.
Practice mixed and advanced integration problems.
Solve Previous Years' Questions (PYQs).
Finish revision with a chapter-wise mock test.

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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Class 12 Maths Chapter 7: Integrals

Integration is one of the fundamental concepts of Calculus and serves as the inverse process of differentiation. The Integrals chapter introduces techniques for evaluating indefinite and definite integrals and explains their applications in finding areas under curves, accumulated quantities, and solving mathematical problems. It forms the foundation for advanced topics such as Applications of Integrals and Differential Equations, making it an essential chapter for CBSE Board, JEE Main, and JEE Advanced.

At ALLEN, Class 12 Maths Chapter 7: Integrals study materials are designed to simplify Integrals through concise theory, important formulas, standard integration techniques, solved examples, and exam-oriented tips. Whether you're revising before board exams or preparing for competitive entrance examinations, these notes provide a structured approach for mastering the chapter efficiently.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	4–5 Hours
Key Concepts Covered	Integration by Substitution, Integration by Parts, Partial Fractions, Indefinite Integrals, Definite Integrals

2.0Chapter 7 Smart Study Material by ALLEN Experts: Download PDFsResource

Status
NCERT Solutions	Available Soon
Important Questions	Available Soon
Formula Sheet PDF	Available Soon
Previous Years' Questions (PYQs)	Available Soon
Mock Test	Available Soon
Practice Worksheets	Available Soon
Quick Revision Notes PDF	Available Soon
Mind Maps	Available Soon

Integral Calculus

Integral Calculus is the branch of calculus focused on accumulation, such as finding areas under curves, volumes, and total changes. It involves indefinite integrals (antiderivatives) and definite integrals (numerical values over intervals). Integral calculus essentially reverses differentiation, helping us reconstruct a function from its rate of change. Widely used in physics, engineering, and economics, it allows us to analyze continuous change and solve real-world problems involving motion, growth, and distribution.

1.0What is Integral Calculus?

Integral Calculus is the branch of calculus concerned with accumulation—how small pieces add up to make a whole. If differential calculus is about breaking things down (derivatives), then integral calculus is about putting things back together (integrals).

It primarily deals with:

Indefinite Integrals: Represent antiderivatives of functions.
Definite Integrals: Represent actual values, like area or total change.

Related Video:

2.0Indefinite Integrals

An indefinite integral is the reverse process of differentiation. It gives a general form of all antiderivatives of a function and includes an arbitrary constant of integration C, since derivatives of constants are zero.

$\int f (x) d x = F (x) + C$

Here:

f(x) is the integrand,
F(x) is any function such that F'(x) = f(x),
C is the constant of integration.

Example:

$\int 2 x d x = x^{2} + C$

Indefinite integrals are functions, not numbers. They're used when you're looking for a general formula rather than a specific value.

3.0Definite Integrals

A definite integral gives the net accumulation of a quantity over an interval [a, b]. It produces a numerical value and has direct geometric meaning—like the area under a curve.

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

Where:

F(x) is the antiderivative of f(x),
a and b are the limits of integration.

Key Properties:

$\int_{a}^{a} f (x) d x = 0$

$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$

If f(x) $\geq$ 0, the integral represents area.

Example:

$\int_{0}^{2} x^{2} d x = [\frac{x ^{3}}{3}]_{0}^{2} = \frac{8}{3}$

Definite integrals are numbers, often used to compute areas, volumes, and total change.

1.0The Core Idea: Summing Infinitesimally Small Quantities

To find the area under a curve y = f(x), we slice it into thin rectangles and add their areas. As we take more slices, we get closer to the true area. This infinite sum leads us to a definite integral:

$\int_{a}^{b} f (x) d x$

Where:

f(x): the function
a, b: the limits of integration
dx: an infinitesimally small width

2.0The Fundamental Theorem of Calculus

This theorem connects differentiation and integration:

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

Here, F(x) is an antiderivative of f(x), meaning F'(x) = f(x).

3.0Common Integration Techniques

1. Substitution (u-substitution):
Use when an integral looks like a chain rule in reverse.

$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$

2. Integration by Parts:

$\int u d v = uv - \int v d u$

3. Partial Fractions:
Break complex rational expressions into simpler terms.

4. Trigonometric Identities:
Useful for integrating sin²x, cos²x, etc.

5. Special Integrals:
Learn standard forms like:

$\int \frac{1}{1 + x ^{2}} d x = tan^{- 1} x + C$

4.0Applications of Integral Calculus

Area under curves
Volume of solids of revolution
Displacement and distance from velocity
Center of mass and moment of inertia
Probability distributions (in stats)

5.0Solved Examples on Integral Calculus

Example 1: Evaluate: $\int \frac{x}{1 + x ^{2}} d x$

Solution:
Let u = $1 + x^{2} \Rightarrow d u = 2 x d x$

So,

$\frac{d u}{2} = x d x$

$\int \frac{x}{1 + x ^{2}} d x = \int \frac{1}{u} \cdot \frac{d u}{2} = \frac{1}{2} \int u^{- 1/2} d u$

$= \frac{1}{2} \cdot \frac{u ^{1/2}}{1/2} = u = 1 + x^{2} + C$

Example 2: Evaluate: $\int_{0}^{π} \frac{x s i n x}{1 + c o s ^{2} x} d x$

Solution:

Use the property:

$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$

Let:

$I = \int_{0}^{π} \frac{x s i n x}{1 + c o s ^{2} x} d x$

Apply the property:

$I = \int_{0}^{π} \frac{( π - x ) s i n x}{1 + c o s ^{2} x} d x$

Add both:

$2 I = \int_{0}^{π} \frac{x s i n x + ( π - x ) s i n x}{1 + c o s ^{2} x} d x = \int_{0}^{π} \frac{π s i n x}{1 + c o s ^{2} x} d x$

$2 I = π \int_{0}^{π} \frac{s i n x}{1 + c o s ^{2} x} d x$

Let $u = cos x \Rightarrow d u = - sin x d x$

Change limits:
When x = 0 $\Rightarrow u = 1,$
When $x = π \Rightarrow u = - 1$

So:

$2 I = - π \int_{1}^{- 1} \frac{1}{1 + u ^{2}} d u = π \int_{- 1}^{1} \frac{1}{1 + u ^{2}} d u$

This is a standard integral:

$π [tan^{- 1} u]_{- 1}^{1} = π (tan^{- 1} 1 - tan^{- 1} (- 1)) = π (\frac{π}{4} + \frac{π}{4}) = π \cdot \frac{π}{2} \Rightarrow 2 I = \frac{π ^{2}}{2} \Rightarrow I = \frac{π ^{2}}{4}$

Example 3: Evaluate: $\int_{0}^{3} [x] d x (where [x] is the greatest integer function)$

Solution:

Break it at integers:

$\int_{0}^{3} [x] d x = \int_{0}^{1} 0 d x + \int_{1}^{2} 1 d x + \int_{2}^{3} 2 d x$ = 0 + 1 + 2 = 3

Example 4: Evaluate:

$\int x ln x d x$

Solution:

Use integration by parts:
Let u = $ln x, d v = x d x$
Then $d u = \frac{1}{x} d x, v = \frac{x ^{2}}{2}$

$\int x ln x d x = \frac{x ^{2}}{2} ln x - \int \frac{x ^{2}}{2} \cdot \frac{1}{x} d x = \frac{x ^{2}}{2} ln x - \int \frac{x}{2} d x = \frac{x ^{2}}{2} ln x - \frac{x ^{2}}{4} + C$

6.0Practice Questions

1. Find the integral: $\int (3 x^{2} + 2 x + 1) d x$

2. Evaluate: $\int_{0}^{2} x^{2} d x$

3. Find the area under the curve $y = sin x from x = 0 to x = π .$

4. Solve using substitution: $\int x x^{2} + 1 d x$

5. Evaluate using integration by parts: $\int x e^{x} d x$

Also Read:

Integrals of Particular Functions	Application of Integrals	Integration by Substitution
Integrating Factor	Differentiation and Integration	Integration by Parts
important integration formulas for jee	Integration by Partial Fractions	definite and indefinite integration

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of indefinite and definite integrals.
Apply standard integration formulas and properties.
Solve integrals using substitution, partial fractions, and integration by parts.
Evaluate definite integrals using their properties.
Simplify complex integration problems using appropriate techniques.
Build a strong foundation for Applications of Integrals and Differential Equations.
Solve JEE Main, JEE Advanced, and CBSE Board questions with confidence.

2.05-Minute Quick Revision

Must-Revise Concepts

Indefinite Integrals
Definite Integrals
Fundamental Theorem of Calculus
Standard Integration Formulas
Integration by Substitution
Integration by Parts
Partial Fractions
Properties of Definite Integrals
Trigonometric Integrals
Rational and Irrational Functions

Important Formula Recall

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
$\int e^{x} d x = e^{x} + C$
$\int sin x d x = - cos x + C$
$\int cos x d x = sin x + C$
$\int_{a}^{b} f (x), d x = - \int_{b}^{a} f (x), d x$
$\int_{a}^{a} f (x), d x = 0$

Important Properties / Shortcut Tips

Always identify the appropriate integration technique before solving.
Use substitution when the integrand contains a composite function.
Apply integration by parts for products of functions.
Learn symmetry properties of definite integrals to reduce calculations.
Remember to include the constant of integration in indefinite integrals.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Standard Integration Formulas	High	Easy–Moderate
Integration by Substitution	Very High	Moderate
Integration by Parts	Very High	Moderate
Partial Fractions	High	Moderate
Definite Integrals & Properties	Very High	Moderate–High
Trigonometric Integrals	High	Moderate
Mixed Integration Problems	Very High	High

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Forgetting the constant of integration	Add + C in every indefinite integral.
Choosing the wrong integration method	Identify whether substitution, parts, or partial fractions is suitable.
Incorrect limits in definite integrals	Substitute the upper and lower limits carefully after integration.
Sign errors in trigonometric integrals	Memorize standard integral formulas thoroughly.
Ignoring definite integral properties	Use symmetry and interval properties to simplify lengthy calculations.
Algebraic simplification mistakes	Simplify the integrand before applying integration techniques.
Confusing differentiation and integration formulas	Revise both concepts together for better accuracy.

6.0ALLEN Faculty Tips

Memorize standard integration formulas before attempting advanced problems.
Practice identifying the most suitable integration technique.
Revise all important properties of definite integrals regularly.
Focus on substitution and integration by parts, as they are frequently tested.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.
Avoid lengthy calculations by applying symmetry properties whenever possible.

7.0PYQ Trend Analysis

Integrals is one of the highest-weightage chapters in Calculus and is frequently tested in JEE Main, JEE Advanced, and CBSE Board examinations through concept-based and application-oriented questions.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Standard Integration	1–2	Easy–Moderate
Integration by Substitution	1–2	Moderate
Integration by Parts	1–2	Moderate
Partial Fractions	1	Moderate
Definite Integrals	2–3	Moderate–High
Properties of Definite Integrals	1–2	Moderate

8.0Most Common Question Types

Evaluating indefinite integrals using standard formulas.
Solving integrals using substitution and integration by parts.
Evaluating definite integrals using properties.
Integrating rational and trigonometric functions.
Simplifying complex integrals using algebraic manipulation.
Applying partial fraction decomposition.
Solving mixed integration problems.
Concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise all standard integration formulas.
Practice substitution-based integrals.
Solve integration by parts questions.
Revise partial fraction decomposition techniques.
Learn important properties of definite integrals.
Practice mixed and advanced integration problems.
Solve Previous Years' Questions (PYQs).
Finish revision with a chapter-wise mock test.

Frequently Asked Questions

What is the difference between definite and indefinite integrals?

Why is integration by parts used in solving integrals?

What are the properties of definite integrals?

How are integrals applied in mathematics and real life?

Why is the Integrals chapter important for JEE Main and JEE Advanced?

Join ALLEN!

Class 12 Maths Chapter 7: Integrals

1.0Chapter Snapshot

2.0Chapter 7 Smart Study Material by ALLEN Experts: Download PDFsResource

On this page

Integral Calculus

1.0What is Integral Calculus?

2.0Indefinite Integrals

3.0Definite Integrals

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

What is the difference between definite and indefinite integrals?

Why is integration by parts used in solving integrals?

What are the properties of definite integrals?

How are integrals applied in mathematics and real life?

Why is the Integrals chapter important for JEE Main and JEE Advanced?

Join ALLEN!

Class 12 Maths Chapter 7: Integrals

1.0Chapter Snapshot

2.0Chapter 7 Smart Study Material by ALLEN Experts: Download PDFsResource

On this page

Integral Calculus

1.0What is Integral Calculus?

2.0Indefinite Integrals

3.0Definite Integrals

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page