NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Miscellaneous Exercise

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Chapter 7 – Integrals Miscellaneous Exercise

1.

Q1 integration of rational algebraic expression using substitution method
 Q1 integration setup with rational expression simplified step-by-step
Q1 final result using substitution and evaluated definite integral

2.

Q2 Class 12 integral involving root and trigonometric simplification
Q2 transformation of trigonometric root expression

3.

Q3 definite integral with exponential and logarithmic terms
 Q3 setup of exponential and logarithmic integration steps

4.

Q4 NCERT problem with trigonometric function and limits
Q4 trigonometric function integrated using substitution

5.

Q5 integration using partial fractions and transformation technique
Q5 partial fraction method applied to rational integral
Q5 final evaluated solution with definite bounds

6.

Q6 Class 12 integral requiring rational simplification and limits
Q6 algebraic simplification and integration technique used
Q6 definite integral answer with limit substitution

7.

Q7 definite integral involving square identity and algebraic terms
Q7 square identity applied for integration setup

8.

Q8 trigonometric integral solved using symmetry rule
 Q8 symmetric property used to simplify definite integral

9.

Q9 rational expression with square root under definite limits
Q9 root and rational terms simplified for integration

10.

Q10 NCERT Class 12 question on inverse trigonometric integration
Q10 inverse trig expression setup for integration

11.

Q11 integration involving sin and cos product with definite limits
Q11 trigonometric product solved step-by-step

12.

Q12 algebraic expression integrated using standard formula
Q12 algebraic terms simplified for definite integral

13.

 Q13 root and trigonometric composite integration under limits
Q13 composite expression simplified using substitution

14.

Q14 rational and logarithmic function with definite bounds
Q14 logarithmic expression handled in rational integration
Q14 Class 12 final answer with simplified terms

15.

Q15 Class 12 integration using substitution and identity
Q15 rational expression rearranged for easy integration

16.

Q16 integration with rational function and interval application
 Q16 integral setup with rational limits simplified

17.

Q17 question involving definite integral of square root form
Q17 root function evaluated using substitution

18.

Q18 trigonometric integral using standard identities
 Q19 algebraic transformation used for integration setup

19.

Q19 NCERT integral requiring algebraic transformation and limits
Class-12 Integrals Miscellaneous Exercise  Question 19 Solution - NCERT Answer
Q19 boxed final value from definite integration

20.

Class-12 Integrals Miscellaneous Exercise Question 20-definite integral using even function property-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 20 even function property applied to simplify expression- NCERT Answer

21.

Class-12 Integrals Miscellaneous Exercise Question 21-problem with trig and log terms integration-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 21 trig-log composite function broken into steps- NCERT Answer
Class-12 Integrals Miscellaneous Exercise  Question 21 solution after simplification- NCERT Answer

22.

Class-12 Integrals Miscellaneous Exercise Question 22-integral with exponential expression and definite limits-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 22 exponential integration handled using substitution- NCERT Answer

23.

Class-12 Integrals Miscellaneous Exercise Question 23-question involving rational simplification-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 23 final step showing answer with bounds- NCERT Answer
Class-12 Integrals Miscellaneous Exercise  Question 1 rational expression simplified to integrate
- NCERT Answer

24.

Class-12 Integrals Miscellaneous Exercise Question 24-integration using change of variables and definite bounds-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 24 substitution applied to rational function integration- NCERT Answer

25.

Class-12 Integrals Miscellaneous Exercise Question 25-trigonometric substitution applied in definite integration-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 25 trigonometric substitution applied for simplification- NCERT Answer

26.

Class-12 Integrals Miscellaneous Exercise Question 26-integral involving rational square expression and symmetry-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 26 rational square expression simplified using symmetry- NCERT Answer
Class-12 Integrals Miscellaneous Exercise  Question 26 final boxed answer of definite integral- NCERT Answer

27.

Class-12 Integrals Miscellaneous Exercise Question 27-definite integral with log and exponential combination-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 27 exponential and log expression broken into steps- NCERT Answer
Class-12 Integrals Miscellaneous Exercise  Question 27 simplified and evaluated definite result- NCERT Answer

28.

Class-12 Integrals Miscellaneous Exercise Question 28-algebraic expression integration under limits-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 28 integral simplified using algebraic technique
- NCERT Answer

29.

Class-12 Integrals Miscellaneous Exercise Question 29-integration using composite trigonometric function-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 29 trigonometric composite function simplified for integration- NCERT Answer

30.

Class-12 Integrals Miscellaneous Exercise Question 30-involving definite integration of rational polynomial-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 30 rational polynomial expression integrated in parts- NCERT Answer
Class-12 Integrals Miscellaneous Exercise  Question 32 Solution - NCERT Answer

31.

Class-12 Integrals Miscellaneous Exercise Question 31-function integration using limits and transformation method-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 31 variable transformation and simplification applied- NCERT Answer

32.

Class-12 Integrals Miscellaneous Exercise Question 32-integral with trigonometric square simplification-NCERT
identity applied to simplify trigonometric square term-Answer 32

33.

Class-12 Integrals Miscellaneous Exercise Question 33 -question with root and rational algebraic combination-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 33 rational and root expression handled stepwise
- NCERT Answer

34.

Class-12 Integrals Miscellaneous Exercise Question 34-integration of complex rational expression with limits-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 34 complex rational expression integrated using substitution- NCERT Answer

35.

Class-12 Integrals Miscellaneous Exercise Question 35-definite integral involving log and identity transformation-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 35 definite integral using logarithmic property
- NCERT Answer

36.

Class-12 Integrals Miscellaneous Exercise Question 36-substitution-based integration of root rational function-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 36 integration of root rational function using substitution- NCERT Answer

37.

Class-12 Integrals Miscellaneous Exercise Question 37-integral using algebraic simplification-NCERT
Class-12 Integrals Miscellaneous Exercise  Question 37 algebraic simplification done before integration- NCERT Answer

Download Miscellaneous Exercise NCERT Solutions PDF

You can download the PDF from the link below for offline study

Class 12 Maths Chapter 7 – Integrals: All Exercises

ExerciseLink
Exercise 7.1View Solutions
Exercise 7.2View Solutions
Exercise 7.3View Solutions
Exercise 7.4View Solutions
Exercise 7.5View Solutions
Exercise 7.6View Solutions
Exercise 7.7View Solutions
Exercise 7.8View Solutions
Exercise 7.9View Solutions
Exercise 7.10View Solutions

Class 12 Integrals- Miscellaneous Exercise Overview

Everything you have thus far discovered about integration is gathered in the Miscellaneous Exercise in Chapter 7 It serves as a sort of last checkpoint where all ideas—definite integrals, indefinite integrals, characteristics of integrals, and application-based problems—combine. This is your opportunity to apply several techniques from past exercises in a mixed form, exactly as you may find in an exam

Integrals Class 12 NCERT Solutions Through a mix of easy and challenging challenges, Miscellaneous Exercise helps students hone their problem-solving techniques. A few issues may combine graph interpretation with definite integrals, some would call for substitution, others for trigonometric identities. This variety gets you ready for competitive assessments like JEE and board exams.

Emphasizing conceptual clarity and practical application of integrals, the 2025 revised syllabus still helps students to develop their ability to be flexible and recognize which integration technique best suits each scenario—a talent quite useful in advanced mathematics and beyond.

Practicing the Integrals Class 12 NCERT Solutions Miscellaneous Exercise helps you not only review but also learn how to handle integrals from all directions. It increases your confidence and improves your capacity for fast and logical solution of invisible problems.

FAQs – Integrals Class 12 Miscellaneous Exercise NCERT

What’s different about the Miscellaneous Exercise compared to previous ones?

It combines all types of integration problems from the chapter, offering a complete revision experience with mixed difficulty levels.

Are there any shortcut techniques for these problems?

While there aren’t direct shortcuts, recognizing identities, symmetry in definite integrals, and graph shapes can speed up the process.

How important is time management for these mixed problems?

Very important. Timed practice with these questions will help you manage exam stress and improve your solving speed.

I feel overwhelmed by the variety of questions. How should I start?

Break it down! Start with familiar question types and slowly try the ones that challenge you. Even 1–2 problems daily can help.

Can I skip this exercise if I’ve done all the previous ones?

Not recommended. This is where your understanding is truly tested — it ensures you’re prepared for unpredictable problem types.

Class 12 Maths Chapters

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

Important Questions for Class 12