NCERT Solutions Class 12 Maths Chapter 7 provides detailed solutions to all 259 questions on Integrals, covering integration by substitution, partial fractions, by parts, and definite integrals. You’ll learn how to evaluate complex integrals using trigonometric identities, apply properties of definite integrals to simplify calculations, and solve area-under-curve problems that frequently appear in CBSE board exams. Each solution includes the exact integration technique to use, common error warnings, and shortcuts for competitive exams.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 7 Integrals – Complete Guide
NCERT Class 12 Chapter 7 – Integrals is one of the most significant chapters in your CBSE Mathematics curriculum, carrying a substantial weightage of 7 marks in the board examination. You will learn that integration is essentially the reverse process of differentiation, often called anti-differentiation, and serves as a powerful tool for finding areas, volumes, and solving real-world problems in physics and engineering.
📊 CBSE Class 12 Maths Chapter 7 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 7 Marks |
| Unit Name | Calculus |
| Difficulty Level | Hard |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
This chapter introduces you to indefinite integrals (primitive functions) and definite integrals (integration with limits). You’ll explore fundamental integration formulas and master various techniques including integration by substitution, integration by parts using the ILATE rule, integration using partial fractions, and special integrals involving trigonometric, exponential, and logarithmic functions. Each method opens new pathways to solve increasingly complex problems that frequently appear in CBSE board examinations.
The practical applications of integrals extend far beyond mathematics. You’ll discover how integration helps calculate areas under curves, distances traveled by moving objects, work done by variable forces, and even probability distributions. Understanding these applications not only strengthens your problem-solving skills but also prepares you for advanced studies in engineering, economics, and physical sciences.
Quick Facts – Class 12 Chapter 7
| 📖 Chapter Number | Chapter 7 |
| 📚 Chapter Name | Integrals |
| ✏️ Total Exercises | 11 Exercises |
| ❓ Total Questions | 259 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Given the high difficulty level and importance of this chapter, expect a mix of 2-mark, 4-mark, and 6-mark questions in your board exam. The chapter builds directly on your knowledge of differentiation from Chapter 5 and connects seamlessly with Chapter 8 on Applications of Integrals. With consistent practice of NCERT solutions and previous year questions, you’ll develop the confidence to tackle any integration problem and secure full marks in this high-weightage chapter.
NCERT Solutions Class 12 Maths Chapter 7 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 7.1 | Complete step-by-step solutions for 22 questions | 📥 Download PDF |
| Exercise 7.2 | Complete step-by-step solutions for 39 questions | 📥 Download PDF |
| Exercise 7.3 | Complete step-by-step solutions for 24 questions | 📥 Download PDF |
| Exercise 7.4 | Complete step-by-step solutions for 23 questions | 📥 Download PDF |
| Exercise 7.5 | Complete step-by-step solutions for 23 questions | 📥 Download PDF |
| Exercise 7.6 | Complete step-by-step solutions for 24 questions | 📥 Download PDF |
| Exercise 7.7 | Complete step-by-step solutions for 11 questions | 📥 Download PDF |
| Exercise 7.8 | Complete step-by-step solutions for 22 questions | 📥 Download PDF |
| Exercise 7.9 | Complete step-by-step solutions for 10 questions | 📥 Download PDF |
| Exercise 7.10 | Complete step-by-step solutions for 21 questions | 📥 Download PDF |
| Exercise 7. Miscellaneous | Complete step-by-step solutions for 40 questions | 📥 Download PDF |
Integrals – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Basic Integral Formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, n \neq -1\) | Integrates a power of x Note: Remember to add the constant of integration ‘C’. The formula is not valid when n = -1. | For integrating polynomial terms like x², x³, etc. |
| Integral of 1/x \(\int \frac{1}{x} \, dx = \ln |x| + C\) | Integrates the reciprocal of x Note: Use absolute value |x| because the logarithm is only defined for positive values. Don’t forget + C | When you have 1/x in the integrand |
| Integral of e^x \(\int e^x \, dx = e^x + C\) | Integrates the exponential function e^x Note: The integral of e^x is itself, plus the constant of integration. | Directly integrate e^x |
| Integral of a^x \(\int a^x \, dx = \frac{a^x}{\ln a} + C\) | Integrates the exponential function a^x Note: Remember to divide by the natural logarithm of the base ‘a’. | When you need to integrate exponential functions where the base is not ‘e’ |
| Integral of sin(x) \(\int \sin x \, dx = -\cos x + C\) | Integrates the sine function Note: The integral of sin(x) is -cos(x), plus the constant of integration. Pay attention to the negative sign! | Directly integrate sin(x) |
| Integral of cos(x) \(\int \cos x \, dx = \sin x + C\) | Integrates the cosine function Note: The integral of cos(x) is sin(x), plus the constant of integration. | Directly integrate cos(x) |
| Integral of tan(x) \(\int \tan x \, dx = \ln |\sec x| + C = -\ln |\cos x| + C\) | Integrates the tangent function Note: Can also be written as -ln|cos x| + C. Use whichever form is more convenient. | When you have tan(x) in the integrand |
| Integral of cot(x) \(\int \cot x \, dx = \ln |\sin x| + C\) | Integrates the cotangent function Note: Remember the absolute value inside the logarithm. | When you have cot(x) in the integrand |
| Integral of sec(x) \(\int \sec x \, dx = \ln |\sec x + \tan x| + C\) | Integrates the secant function Note: A more complex integral, memorize this one carefully. | When you have sec(x) in the integrand |
| Integral of cosec(x) \(\int \csc x \, dx = \ln |\csc x – \cot x| + C\) | Integrates the cosecant function Note: Similar to the integral of sec(x), memorize it. | When you have csc(x) in the integrand |
| Integral of sec²(x) \(\int \sec^2 x \, dx = \tan x + C\) | Integrates the square of the secant function Note: Direct application, remember + C. | When you have sec²(x) in the integrand |
| Integral of cosec²(x) \(\int \csc^2 x \, dx = -\cot x + C\) | Integrates the square of the cosecant function Note: Watch the negative sign! Integral is -cot(x) + C. | When you have csc²(x) in the integrand |
| Integral of sec(x)tan(x) \(\int \sec x \tan x \, dx = \sec x + C\) | Integrates sec(x) * tan(x) Note: Direct application, easy to remember. | When you have sec(x)tan(x) in the integrand |
| Integral of cosec(x)cot(x) \(\int \csc x \cot x \, dx = -\csc x + C\) | Integrates cosec(x) * cot(x) Note: Watch the negative sign! Integral is -csc(x) + C. | When you have csc(x)cot(x) in the integrand |
📚 Related Study Materials – Class 12 Maths Resources
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