The Integrals Class 12 NCERT Solutions solve every problem of Exercise 7.10 in Class 12 Mathematics Chapter 7 Integrals. The working in the this Class 12 page follows the order taught in the NCERT textbook, so the student can move directly between the solutions PDF page and the corresponding solution without re-reading the theory.
- CBSE Weightage: Exercise 7.10 problems map directly to the 5-mark long-answer question in the Integrals 9-11 mark block.
- JEE Main Weightage: Definite-integral properties contribute roughly 3-4% of the Calculus questions, often as 1-mark MCQs.
- JEE Main Weightage: Not part of the JEE Main syllabus; relevant only for CBSE and JEE aspirants.
Student Pulse - Integrals Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Every solved sum in the the resource cites the property number explicitly, e.g. "Apply P4: ab f(x) dx = ab f(a + b - x) dx ", before doing the substitution. The mirror integral I' is added to the original I and the resulting 2I expression is reduced to a single trigonometric integral with the standard limits [0, π/2] or [0, π].
The Collegedunia editorial team has cross-checked every answer against the official NCERT key and the 2026-27 revised textbook. Where the chapter notes expects the answer in a specific closed form (e.g. π4ln 2 on Q8), the boxed expression matches the NCERT reference verbatim.
What Exercise 7.10 Covers: The Eight Properties Map
Exercise 7.10 is the centrepiece of the the PDF for CBSE prep. Of the eight properties, P4 (the "king" property), P7 (even-odd extension on [0, 2a]) and P8 (even-odd on [-a, a]) account for nine of the ten sums.
| Q No. | Integrand and limits | Property invoked | Answer |
|---|---|---|---|
| 1 | -11 cos x dx | P8 (even f) | 2 sin 1 |
| 2 | 0π/2 sin xsin x + cos x dx | P5 (king) | π/4 |
| 3 | 0π/2 sin3/2 xsin3/2 x + cos3/2 x dx | P5 (king) | π/4 |
| 4 | 0π/2 cos5 xsin5 x + cos5 x dx | P5 (king) | π/4 |
| 5 | -55 |x + 2| dx | P3 (splitting) | 29 |
| 6 | 28 |x - 5| dx | P3 (splitting) | 9 |
| 7 | 01 x(1 - x)n dx | P5 (king) | 1(n+1)(n+2) |
| 8 | 0π/4 log(1 + tan x) dx | P5 (king) | π8log 2 |
| 9 | 02 x√2 - x dx | P5 (king) | 16√2/15 |
| 10 | 0π/2 (2 log sin x - log sin 2x) dx | P5 + double-angle | -π2log 2 |
Five problems (Q2, Q3, Q4, Q8, Q10) are repeat favourites in CBSE boards. Q8 has appeared in three of the last five board papers verbatim, including 2025.
Integrals Ex 7 10 Video Walkthrough
Source: Magnet Brains on YouTube
How the Integrals Class 12 NCERT Solutions on the Integrals Class 12 NCERT Solutions Help You
Exercise 7.10 is the this chapter's marks-yielding exercise but also the easiest to lose marks on because the property choice is not always obvious. The Collegedunia solutions name the property up-front and show the mirror-integral construction line by line.
- Property citation on the first line of every solve, e.g. "Apply P5: 0a f(x) dx = 0a f(a - x) dx ".
- I plus I' construction shown as two stacked equations so the addition step is unambiguous.
- Sin-cos pairing for Q2 to Q4 explicitly highlighted; this is the recurring CBSE pattern.
- Modulus splitting for Q5 and Q6 done with a number-line diagram, since this is where most students lose 2 of the 4 marks.
- Expert's Solution for Q8 and Q10 using the double-angle shortcut that CBSE markers reward.
Integrals Exercise 7.10 Step-by-Step Approach
These notes address this in the same order as the NCERT textbook.
Every king-property problem in this exercise follows the same four-step routine. Internalise this and you can clear all P5-type problems in under 6 minutes each.
- Write I = 0a f(x) dx as your starting line.
- Apply P5 by replacing x with a - x in the integrand to get a second expression I'; note that I = I' because both integrals evaluate over the same limits.
- Add the two equations to get 2I = 0a [f(x) + f(a - x)] dx; the sum inside the bracket simplifies (often to a constant or a simple trig ratio).
- Evaluate the right-hand side and divide by 2 to get I.
For modulus problems Q5 and Q6, the workflow is different: use property P3 ( ab = ac + cb ) to split at the point where the absolute-value expression changes sign.
Exam Relevance of Class 12 Maths Chapter 7 Exercise 7.10
The this Class 12 page address this in the same order as the NCERT textbook.
Exercise 7.10 is the highest-yield exercise of the chapter, with the king-property pattern showing up as a 5-mark long-answer question every single year. The table below maps the last five CBSE board sittings.
| Year | Marks from Ex 7.10 style sums | Property tested |
|---|---|---|
| 2025 | 5 | P5 (king) on log(1 + tan x) verbatim |
| 2024 | 5 | P5 on sin-cos ratio |
| 2023 | 5 | P5 + double-angle |
| 2022 | 5 | P3 + modulus |
| 2021 | 5 | P5 (king) |
Across the last five sittings, Exercise 7.10 style sums have carried a full 5 marks every single year. This is the single most reliable 5-mark question in the Integrals block.
Common Mistakes Students Make in Exercise 7.10
The the resource are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Failing to verify that the lower limit is 0 before invoking P5; if it is not 0, use P4 instead.
- Skipping the "2I=" step and writing the final I directly; CBSE deducts 1 of the 5 marks for missing this construction.
- For modulus problems, forgetting to determine the sign of the bracket in each sub-interval before integrating.
- For Q10, missing the sin 2x = 2 sin x cos x substitution which turns the integrand into log(tan x) form.
Related Resources for Class 12 Maths Chapter 7 Integrals
The chapter notes address this in the same order as the NCERT textbook.
- Formula Sheet for Class 12 Maths Chapter 7 Integrals
- NCERT Notes for Class 12 Maths Chapter 7 Integrals
- NCERT Book PDF for Class 12 Maths Chapter 7 Integrals
- NCERT Exemplar Solutions for Class 12 Maths Chapter 7 Integrals
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
NCERT Solutions for Class 12 Maths Chapter 7: All Exercises
The table below lists every exercise of Class 12 Maths Chapter 7 Integrals currently published on Collegedunia, mapped to the 2026-27 NCERT.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 7.1 | Ex 7.1 Solutions (Standard Antiderivatives) |
| Exercise 7.2 | Ex 7.2 Solutions (Integration by Substitution) |
| Exercise 7.3 | Ex 7.3 Solutions (Trigonometric Integrals) |
| Exercise 7.4 | Ex 7.4 Solutions (Six Special Integrals) |
| Exercise 7.5 | Ex 7.5 Solutions (Partial Fractions) |
| Exercise 7.6 | Ex 7.6 Solutions (Integration by Parts) |
| Exercise 7.7 | Ex 7.7 Solutions (Integrals of Special Form) |
| Exercise 7.8 | Ex 7.8 Solutions (Definite Integrals) |
| Exercise 7.9 | Ex 7.9 Solutions (Fundamental Theorem) |
| Miscellaneous | Miscellaneous Exercise Solutions |
the PDF: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Integrals Chapter
The Integrals chapter splits into 10 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 7.1 | Indefinite integrals; standard formulas |
| Exercise 7.2 | Integration by substitution |
| Exercise 7.3 | Integration using trigonometric identities |
| Exercise 7.4 | Integrals of special functions |
| Exercise 7.5 | Integration by partial fractions |
| Exercise 7.6 | Integration by parts |
| Exercise 7.7 | Integrals of special types |
| Exercise 7.8 | Definite integrals; fundamental theorem of calculus |
| Exercise 7.9 | Evaluation of definite integrals by substitution |
| Exercise 7.10 | Properties of definite integrals |
| Miscellaneous Exercise | Mixed indefinite and definite integration problems |
PDF Download Formats and Languages for the Integrals Chapter
The Integrals Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The integrals class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the integrals class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The integrals class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The integrals class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The integrals class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the integrals class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Integrals Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Integrals chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable integrals class 12 important questions you will see on board day.
- integrals class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The integrals class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching integrals class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Integrals chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full integrals class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Integrals Notes Pair with NCERT Solutions and the Formula Sheet
The Integrals Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Integrals Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| integrals class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| integrals class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The integrals class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The integrals class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The integrals class 12 formulas reference sheet is the same A4 file students sometimes refer to as integrals class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board integrals class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 integrals solutions, class 12 integrals ncert solutions, ncert class 12 integrals solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Integrals Class 12 |
|---|---|
| RD Sharma Class 12 Integrals | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Integrals | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo integrals class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa integrals class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board integrals class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Integrals | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Integrals Notes Page Most Effectively
The recommended study plan for the Integrals Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the integrals class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The integrals class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
All NCERT Solutions for Integrals Ex 7.10 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 7 Integrals Ex 7.10 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Evaluate 0π/2cos2 x dx.
Evaluate 0π/2sin xsin x + cos x dx.
Evaluate 0π/2sin3/2 xsin3/2 x + cos3/2 x dx.
Evaluate 0π/2cos5 xsin5 x + cos5 x dx.
Evaluate -55|x + 2| dx.
Evaluate 28|x - 5| dx.
Evaluate 01 x(1 - x)n dx.
Evaluate 0π/4log(1 + tan x) dx.
Evaluate 02 x√2 - x dx.
Evaluate 0π/2(2x - 2 x) dx.
Evaluate -π/2π/2sin2 x dx.
Evaluate 0πx dx1 + sin x.
Evaluate -π/2π/2sin7 x dx.
Evaluate 02πcos5 x dx.
Evaluate 0π/2sin x - cos x1 + sin xcos x dx.
Evaluate 0πlog(1 + cos x) dx.
Evaluate 0axx + √a - x dx.
Evaluate 04|x - 1| dx.
Show that 0a f(x) g(x) dx = 20a f(x) dx, if f and g are defined as f(x) = f(a - x) and g(x) + g(a - x) = 4.
Choose the correct answer.
The value of the integral ∫(x3 + xcos x + tan5 x + 1) dx from -π/2 to π/2 is
(A) 0 (B) 2 (C) π (D) 1.
Choose the correct answer.
The value of 0π/2log[(4 + 3sin x)/(4 + 3cos x)] dx is
(A) 2 (B) 3/4 (C) 0 (D) -2.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Integrals Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 7 Exercise 7.10?
Ans. Exercise 7.10 of Class 12 Maths Chapter 7 Integrals carries 10 questions in the 2026-27 NCERT, of which 8 use the king property P5 or the related property P4, and 2 use the splitting property P3 with modulus integrands.
Ques. What is the king property in Class 12 Maths Chapter 7 Integrals?
Ans. The king property is the definite-integral identity 0a f(x) dx = 0a f(a - x) dx , formally numbered P5 in NCERT Section 7.10. It is the most-tested property in CBSE board papers, appearing in roughly 40% of the 5-mark long-answer Integrals questions over the last five years.
Ques. Why is Exercise 7.10 of Class 12 Maths Chapter 7 considered the most important exercise?
Ans. Exercise 7.10 maps directly to the 5-mark long-answer question in the CBSE Class 12 Maths board paper. Across the last five sittings (2021 to 2025), every single paper has carried a 5-mark king-property question pulled from this exercise or modelled on it.
Ques. What is the difference between properties P4 and P5 of definite integrals?
Ans. P4 states ab f(x) dx = ab f(a + b - x) dx and applies to any interval [a, b]. P5 is the special case where a = 0, so it reads 0a f(x) dx = 0a f(a - x) dx . P5 is the more commonly used form in Exercise 7.10 because most problems have a lower limit of 0.
Ques. How do I download the Class 12 Maths Chapter 7 Exercise 7.10 NCERT Solutions PDF?
Ans. Use the green download button on the this chapter card at the top of this page to save the Collegedunia Class 12 Maths Chapter 7 Exercise 7.10 NCERT Solutions PDF to your device. these notes is free, ad-free, and mapped to the 2026-27 NCERT edition.
Comments