Chapter 7: Integration Class 12 NCERT Solutions PDF | Free Handwritten Notes

Chapter 7: Integration Class 12 NCERT Solutions PDF | Free Handwritten Notes – Integration Class 12 NCERT Solutions PDF): Integration is a fundamental concept in calculus and one of the most important topics in Class 12 Mathematics. Often referred to as the inverse process of differentiation, integration helps us determine a function when its derivative is known. This chapter introduces students to both indefinite integrals (general form without limits) and definite integrals (with upper and lower limits). It covers standard integration techniques such as substitution, integration by parts, and partial fractions.

Understanding integration is essential not only for board exams but also for competitive exams like JEE, NEET, and various entrance tests. It also lays the groundwork for real-world applications like calculating area under curves, velocity from acceleration, and solving differential equations. By mastering this chapter, students gain powerful tools for mathematical modeling and analysis. The structured approach in the NCERT textbook ensures conceptual clarity and prepares students for both academic and practical problem-solving.

Preview of Chapter 7: Integration – Class 12 NCERT Solutions | Free PDF Download | Handwritten Notes Part -1

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Preview of Chapter 7: Integration – Class 12 NCERT Solutions | Free PDF Download | Handwritten Notes Part -2

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📘 Definition of Integration (Class 12) Chapter 7 Math’s

Integration is the inverse process of differentiation. It is a mathematical method used to find a function when its derivative is known. In other words, if ddx[F(x)]=f(x),\frac{d}{dx}[F(x)] = f(x),dxd​[F(x)]=f(x),

then integration gives us: ∫f(x) dx=F(x)+C,\int f(x)\,dx = F(x) + C,∫f(x)dx=F(x)+C,

where:

• ∫\int∫ is the integration symbol
• f(x)f(x)f(x) is the integrand
• dxdxdx indicates integration with respect to x
• F(x)F(x)F(x) is the antiderivative of f(x)f(x)f(x)
• CCC is the constant of integration

Integration helps in calculating areas under curves, total distance from velocity, and solving differential equations. It is mainly classified into two types:

1. Indefinite Integration (without limits)
2. Definite Integration (with limits)

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Key Features of Chapter 7: Integration – Class 12 NCERT Math’s | Free PDF Download | Handwritten Notes

• Subject: Maths (Chapter 7: Integration – Class 12 NCERT )
• Language : Hindi
• Total pages part 1 : 132
• Total pages part 2 : 78
• File size part 1: 88.4 MB
• File size part 2 : 34.4
• Format : PDF
• Well structured and easy to understand
• Includes importance formulas and definitions
• Covers all NCERT syllabus topics
• Useful for quick revision before exam

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📘 Chapter 7: Integration – Important Definitions (Math’s)

Term	Definition
Integration	The process of finding a function from its derivative. It is the inverse operation of differentiation.
Indefinite Integral	An integral that represents a family of functions and contains a constant of integration (C). No limits are involved.
Definite Integral	An integral with upper and lower limits that gives a fixed numerical value representing the area under the curve.
Integrand	The function that is to be integrated, denoted as f(x)f(x)f(x) in ∫f(x)dx\int f(x)dx∫f(x)dx.
Integral	The result or expression obtained after performing integration.
Constant of Integration (C)	An arbitrary constant added to the result of an indefinite integral to represent all possible antiderivatives.
Antiderivative	A function whose derivative is the given function. If F′(x)=f(x)F'(x) = f(x)F′(x)=f(x), then F(x)F(x)F(x) is the antiderivative of f(x)f(x)f(x).
Integration by Substitution	A method of solving integrals by changing variables to simplify the integrand.
Integration by Parts	A method based on the product rule of differentiation, used when the integrand is a product of two functions.
Integration by Partial Fractions	A method used to integrate rational functions by expressing them as a sum of simpler fractions.
Properties of Definite Integral	Rules like additive property, property of limits, and symmetry used to simplify and evaluate definite integrals.

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🎯 How to Prepare for Chapter 7: Integration Math’s – Class 12 NCERT

✅ Step-by-Step Strategy:

Step	What to Do	Tips
1️⃣	Understand Basic Concepts	Grasp the idea of integration as the reverse of differentiation.
2️⃣	Memorize Standard Formulas	Learn all basic and special integrals (like ∫sin⁡x\int \sin x∫sinx, ∫ex\int e^x∫ex, etc.). Use flashcards.
3️⃣	Practice Indefinite Integrals	Start with simple questions, then move to methods: substitution, by parts, and partial fractions.
4️⃣	Solve Definite Integrals	Focus on understanding limits, area under curves, and properties.
5️⃣	Revise Integration Techniques	Practice all three techniques (substitution, parts, partial fractions) daily.
6️⃣	Use NCERT Examples & Exercises	NCERT is most reliable for board exams. Don’t skip solved examples.
7️⃣	Solve Previous Year Questions (PYQs)	Analyze common question patterns and types.
8️⃣	Focus on Application Problems	Practice word problems like area under curves, which are common in exams.
9️⃣	Make a Formula Sheet	Create a quick revision sheet of important formulas.
🔟	Attempt Mock Tests/Timed Practice	Practice under timed conditions to improve speed and accuracy.

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📚 Chapter 7: Integration – Class 12 NCERT Math’s

🧩 Subtopics Table

Sl. No.	Subtopic Name	Description
7.1	Introduction	Basic idea and importance of integration as inverse of differentiation.
7.2	Integration as an Inverse Process of Differentiation	Defines antiderivative and indefinite integrals.
7.3	Methods of Integration	General techniques to solve integrals.
7.3.1	Integration by Substitution	Simplifying the integrand using change of variable.
7.3.2	Integration using Trigonometric Identities	Solving integrals using trig identities like sin⁡2x+cos⁡2x=1\sin^2x + \cos^2x = 1sin2x+cos2x=1.
7.3.3	Integration by Parts	Used when integrand is a product of two functions.
7.3.4	Integration of Some Particular Functions	Standard formulas for exponential, logarithmic, and trigonometric integrals.
7.4	Integration by Partial Fractions	Breaking rational functions into simpler terms for easy integration.
7.5	Definite Integral	Integral with limits; gives area under a curve.
7.6	Fundamental Theorem of Calculus	Links differentiation and integration.
7.7	Evaluation of Definite Integrals by Substitution	Solving definite integrals using substitution method.
7.8	Some Properties of Definite Integrals	Useful properties (like symmetry and limits) to simplify definite integrals.

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✅ Why Are These Handwritten Notes Special? (10 Points)

1. ✍️ Neatly Written & Easy to Read – Clear handwriting with proper spacing helps in better understanding and visual clarity.
2. 📘 Based on NCERT & CBSE Syllabus – 100% aligned with the Class 12 NCERT textbook and latest CBSE exam pattern.
3. 🧠 Concept-Oriented Explanation – Each topic is explained step-by-step to build strong conceptual clarity.
4. 📌 Important Formulas Highlighted – Key formulas and rules are boxed or underlined for quick memorization.
5. 📚 Covers All Methods of Integration – Includes substitution, by parts, partial fractions, and special integrals.
6. 🧾 Includes Board-Focused Questions – Covers previous year questions (PYQs) and expected problems for board exams.
7. 🔁 Quick Revision Friendly – Ideal for last-minute revision; concise but complete.
8. 🎯 Solved Examples Included – Every concept is followed by solved problems for better practice.
9. 📂 Downloadable & Printable – Easily available in PDF format for offline access.
10. 🏆 Created by Experts – Notes are made by toppers and experienced educators with board exam strategy in mind.

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✅ Top 15 Benefits of Using Handwritten Notes 📘 Chapter 7: Integration – Class 12 NCERT

1. ✍️ Better Retention
   Handwriting activates memory — making it easier to remember formulas and steps.
2. 🧠 Concept Clarity
   Notes break down complex integration methods into simple, understandable steps.
3. 🔍 Focused Content
   Only exam-relevant concepts and questions are included — no extra theory.
4. 📘 NCERT-Aligned
   100% syllabus coverage as per NCERT and CBSE board guidelines.
5. 📌 Important Formulas Highlighted
   All essential formulas for indefinite and definite integrals are clearly marked.
6. 🧾 Step-by-Step Solutions
   Each example and method is explained in detailed handwritten steps.
7. 🎯 Board Exam Oriented
   Focus on questions that commonly appear in CBSE and other board exams.
8. 📂 Easy to Revise
   Structured notes make last-minute revision efficient and stress-free.
9. 🧩 Visual Learning
   Diagrams, arrows, highlights, and formatting improve comprehension.
10. 🧾 Covers All Integration Methods
    Includes substitution, parts, partial fractions, trigonometric identities, etc.
11. 🔄 Quick Formula Recap Sheets
    Handy for memorizing standard and special integrals.
12. 💻 Printable PDF Format
    Easy to download, print, and carry — anytime, anywhere.
13. 💬 Student-Friendly Language
    No bookish jargon — explained like a teacher explains on the board.
14. 💡 Tips & Tricks Included
    Smart tricks and shortcuts for faster solving in exams.
15. 📈 Improves Confidence
    Clear understanding and quick revision boost your exam confidence.

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📌 Important Topics – Chapter 7: Integration – Class 12 NCERT

🔢 Sr. No.	📘 Topic Name	📌 Why It’s Important
1️⃣	Integration as the Inverse of Differentiation	Basic concept — foundation of all integration problems.
2️⃣	Standard Integrals	Frequently used in all types of questions.
3️⃣	Integration by Substitution	Common method in both indefinite and definite integrals.
4️⃣	Integration using Trigonometric Identities	Helpful in transforming and simplifying trigonometric integrals.
5️⃣	Integration by Parts	Essential when integrand is a product of two functions.
6️⃣	Integration by Partial Fractions	Used in rational functions — expected in long questions.
7️⃣	Some Special Integrals	Includes forms like 1a2−x2\frac{1}{\sqrt{a^2 – x^2}}a2−x2​1​, 1a2+x2\frac{1}{a^2 + x^2}a2+x21​, etc.
8️⃣	Definite Integrals	High-weightage topic in board exams — area and limit-based questions.
9️⃣	Fundamental Theorem of Calculus	Bridges the link between differentiation and integration.
🔟	Properties of Definite Integrals	Time-saving properties; help simplify complicated expressions.
1️⃣1️⃣	Evaluation of Definite Integrals by Substitution	Useful for board questions where direct evaluation is difficult.
1️⃣2️⃣	Geometrical Interpretation of Integrals	Understanding area under curves — helpful in application-based questions.

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📘 Class 12 Math’s – Chapter 7: Integration 🧮 Important Formulas

🔹 1. Basic Integration Formulas

Function	Integral
∫xndx\int x^n dx∫xndx	xn+1n+1+C\frac{x^{n+1}}{n+1} + Cn+1xn+1​+C (n ≠ -1)
∫1xdx\int \frac{1}{x} dx∫x1​dx	( \ln
∫exdx\int e^x dx∫exdx	ex+Ce^x + Cex+C
∫axdx\int a^x dx∫axdx	axln⁡a+C\frac{a^x}{\ln a} + Clnaax​+C

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🔹 2. Trigonometric Function Integrals

Function	Integral
∫sin⁡xdx\int \sin x dx∫sinxdx	−cos⁡x+C-\cos x + C−cosx+C
∫cos⁡xdx\int \cos x dx∫cosxdx	sin⁡x+C\sin x + Csinx+C
∫sec⁡2xdx\int \sec^2 x dx∫sec2xdx	tan⁡x+C\tan x + Ctanx+C
∫csc⁡2xdx\int \csc^2 x dx∫csc2xdx	−cot⁡x+C-\cot x + C−cotx+C
∫sec⁡xtan⁡xdx\int \sec x \tan x dx∫secxtanxdx	sec⁡x+C\sec x + Csecx+C
∫csc⁡xcot⁡xdx\int \csc x \cot x dx∫cscxcotxdx	−csc⁡x+C-\csc x + C−cscx+C

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🔹 3. Special Integrals

Function	Integral
∫1a2−x2dx\int \frac{1}{\sqrt{a^2 – x^2}} dx∫a2−x2​1​dx	sin⁡−1(xa)+C\sin^{-1} \left( \frac{x}{a} \right) + Csin−1(ax​)+C
∫1a2+x2dx\int \frac{1}{\sqrt{a^2 + x^2}} dx∫a2+x2​1​dx	( \ln \left
∫1a2+x2dx\int \frac{1}{a^2 + x^2} dx∫a2+x21​dx	1atan⁡−1(xa)+C\frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + Ca1​tan−1(ax​)+C
∫1×2−a2dx\int \frac{1}{\sqrt{x^2 – a^2}} dx∫x2−a2​1​dx	( \ln \left

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🔹 4. Integration by Parts

Formula
∫u⋅v dx=u∫v dx−∫(dudx⋅∫v dx)dx\int u \cdot v\,dx = u \int v\,dx – \int \left( \frac{du}{dx} \cdot \int v\,dx \right) dx∫u⋅vdx=u∫vdx−∫(dxdu​⋅∫vdx)dx

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🔹 5. Properties of Definite Integrals

Property	Formula
Reversal of limits	∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx∫ab​f(x)dx=−∫ba​f(x)dx
Same limits	∫aaf(x) dx=0\int_a^a f(x)\,dx = 0∫aa​f(x)dx=0
Additivity over intervals	∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx∫ab​f(x)dx+∫bc​f(x)dx=∫ac​f(x)dx
Even function f(−x)=f(x)f(-x) = f(x)f(−x)=f(x)	∫−aaf(x) dx=2∫0af(x) dx\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx∫−aa​f(x)dx=2∫0a​f(x)dx
Odd function f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x)	∫−aaf(x) dx=0\int_{-a}^{a} f(x)\,dx = 0∫−aa​f(x)dx=0

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FAQs on Class 12 Math’s Handwritten Notes PDF Download Chapter 7: Integration

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Short & Effective Preparation Tips for Class 12 Math’s

🔢 Tip No.	📌 Preparation Tip

1️⃣	Start with NCERT – Complete all exercises & examples thoroughly.

2️⃣	Master the Formulas – Make a separate sheet and revise daily.

3️⃣	Focus on Conceptual Clarity – Understand the why behind each step, not just the how.

4️⃣	Practice Daily – Solve 5–10 questions from each topic consistently.

5️⃣	Revise Weak Areas Regularly – Don’t skip topics you find difficult (like Integration by Parts).

6️⃣	Use Handwritten Notes – Quick to revise and personalized for better retention.

7️⃣	Solve PYQs (Past Year Questions) – Spot patterns and important questions.

8️⃣	Attempt Sample Papers Weekly – Practice under timed conditions to build speed.

9️⃣	Clear Doubts Instantly – Don’t pile up confusion—ask teachers, friends, or use apps.

🔟	Stay Consistent & Confident – Maths needs regular effort, not last-minute cramming.

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Avoid These Common Mistakes in Chapter 7: Integration – Class 12 Math’s

🔢 Mistake	🚫 What Goes Wrong	✅ How to Avoid It
1️⃣ Forgetting + C in indefinite integrals	Leads to loss of marks in board exams	Always add “+ C” at the end unless it’s a definite integral
2️⃣ Confusing definite and indefinite integrals	Students mix up the two and miss applying limits	Learn the difference and apply limits only in definite integrals
3️⃣ Wrong integration method choice	Using by-parts where substitution was easier	Identify the integrand type before choosing method
4️⃣ Errors in u-v selection in Integration by Parts	Choosing incorrect u and v makes the problem tougher	Use LIATE rule: Logarithmic > Inverse > Algebraic > Trig > Exponential
5️⃣ Skipping simplification before integration	Leads to complicated solutions	Always simplify the function before integrating
6️⃣ Not practicing partial fractions properly	Gets stuck in rational function questions	Practice breaking into partial fractions step-by-step
7️⃣ Incorrect substitution limits in definite integrals	Forget to change limits according to the new variable	Change limits if you change the variable during substitution
8️⃣ Misapplying trigonometric identities	Wrong identity = wrong result	Revise standard identities frequently
9️⃣ Poor handling of modulus & square root functions	Forgetting domain restrictions or signs	Always consider domain while integrating under square root/modulus
🔟 Ignoring graphical meaning of definite integrals	Leads to confusion in application-based questions	Understand integration as area under curve
1️⃣1️⃣ Relying only on formulas	No understanding = zero flexibility in new questions	Practice derivations and application-based problems
1️⃣2️⃣ Not revising properties of definite integrals	Missed shortcuts, longer solving time	Memorize and apply properties like symmetry, limit shift, etc.
1️⃣3️⃣ Not labeling steps in board exam	Poor presentation loses marks	Write “By substitution”, “By parts”, etc., clearly
1️⃣4️⃣ Copying limits incorrectly from the question	Wrong answer even if process is correct	Always double-check the limits before solving
1️⃣5️⃣ Ignoring solved examples in NCERT	Misses variety of standard and tricky questions	Solve all solved examples along with exercises

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📊 Summary Table – Chapter 7: Integration – Class 12 Maths

🔢 Topic	📘 Description	🎯 Exam Weightage	📈 Difficulty Level
Integration as Inverse of Differentiation	Basic concept linking integration and differentiation.	🌟 Moderate	⭐ Easy
Standard Integrals	Common formulas like ∫xndx\int x^n dx∫xndx, ∫exdx\int e^x dx∫exdx, ∫sin⁡xdx\int \sin x dx∫sinxdx, etc.	🌟 High	⭐ Easy
Methods of Integration	Techniques: substitution, by parts, partial fractions, trig identities.	🌟 Very High	⭐⭐ Medium
Special Integrals	Integrals involving square roots and inverse trigonometric functions.	🌟 Moderate	⭐⭐ Medium
Definite Integrals	Integral with limits; gives area under the curve.	🌟 Very High	⭐⭐ Medium
Properties of Definite Integrals	Simplifying definite integrals using symmetry, shifting limits, etc.	🌟 High	⭐⭐ Medium
Fundamental Theorem of Calculus	Links definite integrals to antiderivatives.	🌟 Moderate	⭐⭐ Medium
Evaluation using Substitution	Solving definite integrals by changing variable and adjusting limits.	🌟 High	⭐⭐ Medium
Application (Area under curves – intro)	Conceptually important, though detailed applications come in next chapter.	🌟 Low	⭐⭐ Medium
NCERT Exercise Coverage	Exercises 7.1 to 7.11 + Miscellaneous (covering all types of questions).	🌟 Very High	⭐⭐ Medium to High

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🔚 Conclusion – Chapter 7: Integration – Class 12 Maths

Chapter 7: Integration is one of the most crucial and scoring chapters in Class 12 Mathematics. It forms the foundation for higher-level mathematics in competitive exams like JEE, CUET, and future college courses like engineering and economics. The chapter begins by introducing integration as the reverse process of differentiation and gradually builds up through various methods such as substitution, integration by parts, and partial fractions.

It also explains definite integrals, their properties, and the application of definite integrals in calculating area under curves — paving the way for understanding real-world mathematical modeling. A strong grasp of formulas, technique selection, and rigorous practice is the key to mastering this chapter. Regular revision using handwritten notes and solving NCERT + PYQs can help boost both speed and accuracy.

In summary, Integration is not just a chapter but a core mathematical skill. With proper understanding and practice, students can confidently tackle even the most challenging problems in exams.

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