NCERT Solutions for Class 12 Mathematics Chapter 7
NCERT solutions for Class 12 Mathematics Chapter 7 Integrals – complete answers & explanations
This blog provides complete NCERT solutions for Class 12 Mathematics Chapter 7 Integrals. This chapter introduces students to the concept of integration, which is the reverse process of differentiation and an important part of calculus. It helps students learn how to solve different types of integrals including algebraic, trigonometric, exponential, and logarithmic functions. Understanding this chapter is essential for building strong mathematical skills and preparing for board exams. These solutions help students follow the correct approach and understand how to solve problems step by step. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance.
What this NCERT chapter covers?
1. Understanding the concept of integrals
and their importance in calculus
2. Learning how to solve indefinite integrals
3. Applying standard formulas of integration in different problems
4. Working with algebraic expressions in integration
5. Solving trigonometric integrals using identities and formulas
6. Understanding exponential and logarithmic integrals
7. Practicing different types of integration techniques
8. Improving step-by-step problem-solving skills
9. Developing accuracy in calculations and simplifications
10. Building confidence for board exam questions
11. Strengthening overall understanding of higher mathematics
How to use these NCERT solutions?
1. Start by attempting each question
from the worksheet on your own
2. Try to apply formulas and solve step by step before checking answers
3. Use the solutions to verify your final answers
4. Carefully compare your steps with the correct method
5. Identify mistakes and understand where you went wrong
6. Practice similar types of questions again for better clarity
7. Parents and teachers can guide students using these structured solutions
8. Follow the same order as given in the worksheet for better learning
9. Use these solutions during revision before exams
Important tips & tricks for students
1. Always remember and revise important integration formulas
2. Do not forget to add the constant of integration (C)
3. Practice different types of problems regularly
4. Be careful with signs while solving trigonometric functions
5. Double-check calculations to avoid simple mistakes
6. Break complex problems into smaller steps
7. Focus on accuracy along with speed
8. Practice consistently to improve confidence
9. Review mistakes and learn from them
NCERT solutions – complete answer key
Exercise 7.1
1. –1/2 cos 2x + C
2. 1/3 sin 3x + C
3. 1/2 e^(2x) + C
4. (ax + b)³ / (3a) + C
5. –1/2 cos 2x – (4/3)e^(3x) + C
6. e^(x³) + x + C
7. x²/2 – log|x| + C
8. ax³/3 + bx²/2 + cx + C
9. x²e^x + C
10. x²/2 – log|x| + C
11. (5/2)x² + (4/3)x³ – log|x| + C
12. x² + (3/2)x³ + 4 log|x| + C
13. x³/3 + x²/2 + x – log|x| + C
14. x – x²/2 + C
15. x³ + x² + 3x + C
16. 2x² – 3 sin x + e^x + C
17. x² – (3/2)x² – 5 cos x + C
18. –cosec x – cot x + C
19. –tan x – cot x + C
20. tan x – 3 sin x + C
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
Why NCERT solutions help students?
NCERT solutions help students build strong concepts, understand the correct answering approach, and prepare effectively for exams. They improve accuracy, boost confidence, and ensure students follow the expected pattern in board examinations.
Help your child master Class 12 Mathematics concepts with clear and structured NCERT solutions for better exam performance.