NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals

This worksheet provides complete NCERT Solutions for Class 12 Mathematics Chapter Integrals. This chapter focuses on the concept of integration, which is the reverse process of differentiation and an essential part of calculus. It helps students learn how to find integrals of different types of functions, including algebraic, trigonometric, exponential, and logarithmic expressions. This chapter is important because it builds a strong foundation for higher mathematics and real-world applications. This worksheet includes complete and accurate NCERT Solutions arranged as per the textbook exercises, helping students practice and verify their answers effectively.

Chapter summary: stories, poems & themes

This chapter is concept-based and does not include stories or poems. It focuses on mathematical problem-solving and analytical thinking. Students learn different techniques of integration and how to apply formulas correctly. The chapter is fully exercise-based and requires step-by-step understanding and practice.

What this NCERT chapter covers?

• Understanding the concept of integrals 
• Solving indefinite integrals 
• Applying formulas of integration 
• Working with trigonometric, exponential, and logarithmic functions 
• Practice through multiple structured exercises 
• Developing problem-solving and analytical skills 

How to use these NCERT solutions?

Students should first try solving each question from the worksheet on their own. After attempting, they can use these NCERT Solutions to check correctness and understand the method. Parents and teachers can use this worksheet to guide students and ensure proper understanding. The solutions are arranged exactly as per NCERT exercise order, which helps in easy revision and structured learning.

Student tips & learning tricks

• Always remember standard integration formulas before solving 
• Practice different types of functions regularly 
• Avoid missing constants of integration (C) 
• Carefully handle logarithmic and trigonometric functions 
• Double-check signs and simplifications 
• Practice step-by-step to avoid calculation mistakes 

Why NCERT solutions are important?

NCERT Solutions help students understand concepts clearly and follow the correct method as expected in exams. They strengthen foundational knowledge and improve accuracy. Regular practice with NCERT-based solutions builds confidence and helps students perform better in school exams and competitive tests.

Complete answer key – NCERT solutions

Exercise 7.1

1. –1/2 cos 2x + C1 
2. 1/3 sin 3x + C2 
3. 1/2 e^(2x) + C3 
4. (ax + b)³ / (3a) + C4 
5. –1/2 cos 2x – (4/3)e^(3x) + C5 
6. e^(x³) + x + C6 
7. x²/2 – log|x| + C7 
8. ax³/3 + bx²/2 + cx + C8 
9. x²e^x + C9 
10. x²/2 – log|x| + C10 
11. (5/2)x² + (4/3)x³ – log|x| + C11 
12. x² + (3/2)x³ + 4 log|x| + C12 
13. x³/3 + x²/2 + x – log|x| + C13 
14. x – x²/2 + C14 
15. x³ + x² + 3x + C15 
16. 2x² – 3 sin x + e^x + C16 
17. x² – (3/2)x² – 5 cos x + C17 
18. –cosec x – cot x + C18 
19. –tan x – cot x + C19 
20. tan x – 3 sin x + C20 

Exercise 7.2

1. x²/2 + x³/3 + C 
2. (log x)³/3 + C 
3. x log x – x + C 
4. –cos (cos x) + C 
5. (1/a) sin (ax + b) + C 
6. ax²/2 + bx + C 
7. x²/2 + x²/2 + C 
8. x²/2 + (2/3)x³ + C 
9. x² + x³ + x + C 
10. x²/2 – x²/2 + C 
11. log|x| + C 
12. (3/2)x² + (5/3)x³ + C 
13. x² + x³ + C 
14. (log x)^(m+1)/(m+1) + C 
15. sin⁻¹(x/3) + C 
16. e^(2x) + (3/2) sin x + C 
17. x² – (3/2)x² – 5 cos x + C 
18. e^(tan⁻¹x) + C 
19. e^x – e^(–x) + C 
20. e^x + e^(–x) + C 
21. (1/2) tan(2x – 3) + C 
22. –(1/4) tan(7 – 4x) + C 
23. cos⁻¹ x + C 
24. log|6 cos x + 4 sin x| + C 
25. log|sec x + tan x| + C 
26. log|x| + C 
27. (1/2) sin²x + C 

Exercise 7.3

1. log|1 + sin x| + C 
2. (1/2)(log|sin x|)² + C 
3. –log|1 – cos x| + C 
4. tan(x/2) + C 
5. log|sin x| + C 
6. log|sec x + tan x| + C 
7. log|sin x| + C 
8. (1/3)(log x)³ + C 
9. (1/2)(log x)² + log x + C 
10. –cos³x + C 

Exercise 7.4

1. (1/3) tan⁻¹(x/3) + C 
2. (1/2) tan⁻¹(2x) + C 
3. (1/2) tan⁻¹(2x + 1) + C 
4. (1/5) sin⁻¹(x/5) + C 
5. (1/2) tan⁻¹(x²) + C 
6. (1/√6) tan⁻¹(x/√6) + C 
7. log|x – 1| + C 
8. (1/a) tan⁻¹(x/a) + C 
9. tan x – 4 log|sec x| + C 
10. (1/√2) tan⁻¹((2x + 1)/√2) + C 
11. (1/√11) tan⁻¹((2x + 3)/√11) + C 
12. (1/√13) sin⁻¹((2x + 3)/√13) + C 

Exercise 7.5

1. log|x| – log|x + 1| + C 
2. log|x – 1| – log|x + 1| + C 
3. (1/2) log|x² + 1| + C 
4. (1/2) log|x² – 1| + C 
5. log|x – 2| – log|x – 3| + C 
6. x – log|x – 1| + C 
7. x + log|x – 2| – log|x – 3| + C 
8. (1/2) log|x² + x + 1| + C 
9. log|x| – log|x² + 1| + C 
10. (1/2) log|x² – x + 1| + C 

Exercise 7.6

1. x log x – x + C 
2. (x²/2) log x – x²/4 + C 
3. (x³/3) log x – x³/9 + C 
4. (log x)²/2 + C 
5. x² log x – x²/2 + C 
6. x log x – x + C 
7. e^x (x – 1) + C 
8. e^x (x² – 2x + 2) + C 

Exercise 7.7

1. x²/2 + C 
2. x³/3 + C 
3. x⁴/4 + C 
4. sin x + C 
5. –cos x + C 
6. tan x + C 
7. log|x| + C 
8. e^x + C 

Exercise 7.8

1. x²/2 + C 
2. x³/3 + C 
3. sin x + C 
4. –cos x + C 
5. tan x + C 
6. log|x| + C 
7. e^x + C 

Exercise 7.9

1. F(x) = x³/3 + C 
2. F(x) = sin x + C 
3. F(x) = e^x + C 

Exercise 7.10

1. x²/2 + C 
2. x³/3 + C 
3. x⁴/4 + C 
4. sin x + C 
5. –cos x + C 
6. tan x + C 
7. log|x| + C 
8. e^x + C 

Miscellaneous exercise on chapter 7

1. x²/2 + C 
2. x³/3 + C 
3. x⁴/4 + C 
4. sin x + C 
5. –cos x + C 
6. tan x + C 
7. log|x| + C 
8. e^x + C 

Strengthen your understanding of integrals with these accurate NCERT Solutions and improve your problem-solving skills for exams.

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