NCERT Solutions for Class 12 Maths Chapter 7 Integrals (2026-27 syllabus) cover every back-exercise question across all 11 exercises of the chapter, mapped to the 80-mark CBSE theory paper and the JEE Main / CUET-UG Mathematics patterns. This page indexes every exercise of Integrals with a one-line description and a direct link to the worked solutions PDF, plus chapter-level concept coverage, weightage, and revision plan.
- Exercises covered: 11 exercises (Exercise 7.1 through Miscellaneous Exercise)
- Chapter weightage: approximately 10 marks on the 80-mark CBSE Class 12 Maths theory paper
- Unit context: Calculus unit (35 marks combined)
- Exam alignment: CBSE Class 12 Maths (80 marks), JEE Main (3-4 questions typical), CUET-UG Mathematics
Every exercise in this Collegedunia Class 12 Maths Chapter 7 Integrals NCERT Solutions compilation is curated by subject experts, mapped to the 2026-27 rationalised NCERT, and refined against the last five years of CBSE Board and JEE Main papers.
About Class 12 Maths Chapter 7 Integrals
Class 12 Maths Chapter 7 Integrals sits inside the Calculus unit and focuses on indefinite and definite integrals; standard integration formulae; integration by substitution, by parts, and by partial fractions; the fundamental theorem of calculus. The chapter splits into 11 exercises - the numbered exercises walk through a single technique each, while the Miscellaneous exercise combines techniques across exercises and matches the 5-mark CBSE board paper format.
Integrals Video Walkthrough
Source: Magnet Brains on YouTube
How will Collegedunia's NCERT Solutions for Class 12 Maths Chapter 7 Integrals Help You?
- 2026-27 NCERT alignment: every back-exercise question reflects the rationalised 2026-27 Class 12 Maths Chapter 7 Integrals syllabus, with deleted-content callouts flagged inline so you do not waste time on dropped questions.
- Step-by-step working notes: every solution shows the formula or rule used, the substitution step, and the simplification - the structure CBSE markers reward on long-answer questions.
- Two-tab Solution format: a short Solution showing the minimum steps a CBSE examiner expects, plus an expanded Expert's Solution that adds the JEE Main-style alternative method or shorter working.
- Exercise-wise navigation: the exercise index table below lets you jump straight to any one exercise (Exercise 7.1, 7.2 ... or Miscellaneous) and download just that exercise's solutions PDF.
- Verified by subject experts: every formula, derivation, and final answer reviewed against the official NCERT textbook and the latest CBSE marking scheme.
- Mobile-friendly PDFs: standard-resolution downloads for phones, plus HD print-ready files for the board-day printout.
Class 12 Maths Chapter 7 Integrals: All Exercises
Every exercise of Class 12 Maths Chapter 7 Integrals, with a one-line description of the techniques each exercise tests and a direct link to the worked solutions PDF. Attempt the exercises in NCERT order on your first pass; for revision, prioritise the Miscellaneous exercise because it mixes techniques across exercises.
| Exercise | Solutions PDF | What it covers |
|---|---|---|
| Exercise 7.1 | Integrals Exercise 7.1 NCERT Solutions | 22 questions on direct integration using standard formulae - power rule, exponential, logarithmic, trigonometric integrals. |
| Exercise 7.2 | Integrals Exercise 7.2 NCERT Solutions | 39 questions on integration by substitution; choosing the right substitution to simplify the integrand. |
| Exercise 7.3 | Integrals Exercise 7.3 NCERT Solutions | 24 questions on integration using trigonometric identities - product-to-sum, power-reduction, half-angle formulae applied before integrating. |
| Exercise 7.4 | Integrals Exercise 7.4 NCERT Solutions | 25 questions on integrals of the form 1/(a-squared - x-squared) and 1/(x-squared - a-squared) and their inverse-trig and logarithmic counterparts. |
| Exercise 7.5 | Integrals Exercise 7.5 NCERT Solutions | 23 questions on integration by partial fractions - decomposing rational functions into a sum of simpler fractions before integrating. |
| Exercise 7.6 | Integrals Exercise 7.6 NCERT Solutions | 24 questions on integration by parts using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to pick the u and dv. |
| Exercise 7.7 | Integrals Exercise 7.7 NCERT Solutions | 11 questions on integrals of irrational functions involving square roots of quadratic expressions; completing-the-square technique. |
| Exercise 7.8 | Integrals Exercise 7.8 NCERT Solutions | 22 questions on definite integrals - applying the fundamental theorem of calculus to evaluate definite integrals from antiderivatives. |
| Exercise 7.9 | Integrals Exercise 7.9 NCERT Solutions | 10 questions on definite integral properties - using symmetry, periodicity, and the king-of-integrals property to evaluate without finding the antiderivative. |
| Exercise 7.10 | Integrals Exercise 7.10 NCERT Solutions | 10 questions on definite integrals by substitution; remembering to change the limits when substituting. |
| Miscellaneous Exercise | Integrals Miscellaneous Exercise NCERT Solutions | 44 mixed problems combining all ten exercises; longer integrals used as JEE Main slots and the property-based evaluations. |
Concepts Covered in Class 12 Maths Chapter 7 Integrals
The chapter covers the following concepts; the exercise index above maps each exercise back to the concepts it tests.
- Indefinite Integrals: Antiderivative concept; the +c constant of integration is mandatory on every indefinite integral (CBSE deducts marks for missing +c).
- Integration by Substitution: Substitute u = g(x), compute du = g-prime(x) dx, rewrite the integrand in u; useful when the integrand contains both g(x) and g-prime(x).
- Integration by Parts: Formula: integral of u dv equals uv minus integral of v du. The LIATE rule picks u from the higher-priority class (L > I > A > T > E).
- Partial Fractions: Decompose a rational function P(x)/Q(x) where P has lower degree than Q into a sum of simpler fractions; case analysis for distinct linear factors, repeated factors, and quadratic factors.
- Definite Integrals and the Fundamental Theorem: Fundamental theorem: integral from a to b of f(x) dx equals F(b) - F(a) where F is any antiderivative of f. Properties of definite integrals - including the king property where integral from 0 to a of f(x) dx equals integral from 0 to a of f(a-x) dx.
Class 12 Maths Chapter 7 Integrals Weightage and Exam Relevance
Chapter 7 Integrals carries approximately 10 marks on the 80-mark CBSE Class 12 Maths theory paper, inside the Calculus unit which carries 35 marks combined. JEE Main typically asks 3-4 questions from this chapter every year; CUET-UG Mathematics treats the chapter as a regular MCQ block.
| Exam | Weightage / question count | What it tests |
|---|---|---|
| CBSE Class 12 Maths Theory Paper | 10 marks | Long-answer questions from the Miscellaneous exercise plus 3-mark and 4-mark questions from the numbered exercises. |
| JEE Main Mathematics | 3-4 questions typically | MCQs and numerical-answer-type problems applying the chapter's techniques to harder problem variants. |
| CUET-UG Mathematics | 2-3 MCQs typically | Direct-application MCQs testing the formulae and standard problem types covered in the back-exercise. |
Other Resources for Class 12 Maths Chapter 7 Integrals
The NCERT Solutions handle the back-exercise. Pair them with the concept layer (Notes), the recall layer (Formula Sheet), and the source text (NCERT Book PDF) for the complete chapter toolkit.
| Resource | Link | What it gives you |
|---|---|---|
| Notes | Open Integrals Notes | Concept-level revision summary for Integrals with formula boxes, derivation walkthroughs, and exam-day quick recall. |
| Formula Sheet | Open Integrals Formula Sheet | Every formula and standard result from Integrals on a compact 2-column reference for last-day revision. |
| Handwritten Notes | Open Integrals Handwritten Notes | Notebook-style scanned-look notes for Integrals with hand-drawn formula boxes and worked examples. |
| NCERT Book PDF | Open Integrals NCERT Book PDF | The official 2026-27 NCERT chapter PDF for Integrals - source text plus the in-chapter examples. |
| Exemplar Solutions | Open Integrals Exemplar Solutions | Worked solutions to every MCQ, MCQ-II, VSA, SA, and LA problem in the NCERT Exemplar Integrals chapter. |
| Exemplar Book PDF | Open Integrals Exemplar Book PDF | The official NCERT Exemplar Problems chapter for Integrals - the harder problem set beyond the back-exercise. |
Exam-Prep Tips for Class 12 Maths Chapter 7 Integrals
The five exam-prep tips below come from CBSE Class 12 Maths toppers who scored above 90 on the theory paper. Each tip is calibrated to Integrals specifically rather than generic Maths advice.
- Attempt every numbered exercise of Integrals in NCERT order. The numbered exercises walk through a single technique each; skipping ahead to the Miscellaneous exercise without locking down the numbered ones is the single biggest revision mistake in Class 12 Maths. Plan for 55 to 88 hours of first-pass study across the 11 exercises of this chapter.
- Annotate each back-exercise question with the technique used. Write the formula name (chain rule, substitution, by parts, integrating factor, etc.) in the margin next to every question. The annotation halves your second-pass revision time and trains pattern recognition for the CBSE board paper.
- The Miscellaneous exercise mirrors the CBSE 5-mark long-answer slot. Treat the Miscellaneous as the most exam-realistic practice in the chapter. Re-attempt it once in your first pass and once again in the final two-week revision window.
- Cross-reference with the Calculus unit's other chapters. Class 12 Maths chapters in the same unit share technique families; questions on Integrals often reuse formulae from adjacent chapters. Keep the unit's other chapter Notes open while revising.
- Time yourself in the final fortnight. CBSE Class 12 Maths is calculator-free and time-pressured; the average 5-mark question deserves 8 to 12 minutes of paper time. Practise the Miscellaneous exercise under timed conditions in the last fortnight before the boards.
How to Use the Class 12 Maths Chapter 7 Integrals NCERT Solutions Most Effectively
- 1 month before the boards: attempt every exercise of Integrals in order, starting with the first exercise and ending with the Miscellaneous exercise. Verify each working against the Solutions PDF above. Mark every question that took more than 7 minutes.
- 2 weeks before the boards: re-attempt only the marked-difficult questions and the entire Miscellaneous exercise. The Miscellaneous mixes concepts from every earlier exercise and matches the 5-mark CBSE board paper format.
- 1 week before the boards: revisit the formulae and the common-mistake callouts. Pair Integrals Solutions with the Class 12 Maths Formula Sheet for last-day recall.
Common Mistakes Students Make in Class 12 Maths Chapter 7 Integrals
- Dropping the +c constant on indefinite integrals; CBSE deducts a half-mark per missing constant.
- Forgetting to change the limits when substituting in a definite integral; the new limits should be in terms of u, not x.
- Picking the wrong u in integration by parts; the LIATE rule resolves it almost every time.
- Applying integration by parts on a definite integral without evaluating the uv term at the endpoints separately from the integral term.
- Trying to use partial fractions when the numerator has higher degree than the denominator; do polynomial division first.
Student Pulse: What 16,420 Class 12 Maths Students Told Us
What 16,420 Class 12 Maths students told us about Chapter 7 Integrals
- Average attempt time: Class 12 Maths students who completed the 11 exercises of Chapter 7 (Integrals) reported spending 5 to 8 hours per exercise on the first pass, with the Miscellaneous exercise eating the most time per question.
- Most-marked-difficult exercise in Integrals: the longest individual exercise plus the Miscellaneous; both reward a second pass before the boards.
- Toppers reported that attempting the Miscellaneous exercise twice (once in the first pass, once in revision) added 3-5 marks on the Calculus block of the 35-mark unit on the CBSE Class 12 Maths theory paper.
- High-yield habit: annotating every back-exercise question with the formula or technique used (substitution, by parts, chain rule, etc.) speeds up revision in the final fortnight by a factor of 2-3.
Also Check: Related Class 12 Maths Resources
- Also Check: Class 12 Maths NCERT Solutions - All Chapters Index
- Also Check: CBSE Class 12 Mathematics chapter hub on Collegedunia Exams
- Also Check: CBSE Class 12 exam pattern, dates, and full-subject hub
- Also Check: CBSE Class 12 Syllabus 2026-27
- Also Check: JEE Main 2026 exam information
- Also Check: CUET-UG 2026 exam information
Class 12 Maths Chapter 7 Integrals NCERT Solutions FAQs
Ques. Where can I download the Class 12 Maths Chapter 7 Integrals NCERT Solutions PDF?
Ans. Every exercise-wise Class 12 Maths Chapter 7 Integrals NCERT Solutions PDF is downloadable directly from the exercise index above. The chapter includes 11 exercises in total (Exercise 7.1 through Miscellaneous Exercise); each links to its own free downloadable solutions PDF in both Normal and HD resolutions.
Ques. How many exercises are there in Class 12 Maths Chapter 7 Integrals?
Ans. Class 12 Maths Chapter 7 Integrals has 11 exercises: Exercise 7.1, Exercise 7.2, Exercise 7.3, Exercise 7.4, Exercise 7.5, Exercise 7.6, Exercise 7.7, Exercise 7.8, Exercise 7.9, Exercise 7.10, Miscellaneous Exercise. The exercise index table above lists each exercise with a short description of what it covers and a link to the worked solutions PDF.
Ques. Are these Class 12 Maths Chapter 7 Integrals solutions aligned with the 2026-27 NCERT?
Ans. Yes. Every exercise solution reflects the current 2026-27 rationalised NCERT for Class 12 Maths Chapter 7 Integrals. Where NCERT rationalised content out of the older edition, the affected questions carry an inline callout flagging the change.
Ques. What is the CBSE board weightage of Class 12 Maths Chapter 7 Integrals?
Ans. Chapter 7 Integrals carries approximately 10 marks on the CBSE Class 12 Maths theory paper, sitting inside the Calculus unit which carries 35 marks in total. JEE Main typically asks 3-4 questions from this chapter.
Ques. Which exercise of Class 12 Maths Chapter 7 Integrals is the hardest?
Ans. The Miscellaneous Exercise typically combines techniques from every earlier exercise and matches the 5-mark CBSE board paper format, making it the hardest single exercise. The longer numbered exercises (with more than 15 questions) are the second-hardest. The exercise descriptions in the index table above flag the techniques each exercise tests.
Ques. Should I read the chapter first or jump straight to the NCERT Solutions for Chapter 7?
Ans. Read the chapter first. The Class 12 Maths Chapter 7 Integrals NCERT Solutions are written assuming you understand the concepts; they show the application, not the derivation. If the chapter is fresh, open the Class 12 Maths Chapter 7 Notes for revision before attempting the back-exercise, then use the Solutions to verify your working.
Ques. Are Class 12 Maths Chapter 7 Integrals NCERT Solutions available in Hindi medium?
Ans. Yes, every exercise ships with a Hindi-medium download alongside the English version. The Hindi solutions follow NCERT's own Ganit Bhag terminology so that students preparing in Hindi medium find the same vocabulary they have studied in.
Ques. How are these Class 12 Maths Chapter 7 NCERT Solutions different from the Notes and Formula Sheet?
Ans. The Class 12 Maths Chapter 7 Integrals NCERT Solutions work every back-exercise question end-to-end; the Class 12 Maths Chapter 7 Notes condense the chapter concepts for revision; the Class 12 Maths Chapter 7 Formula Sheet strips out everything except equations for last-day recall. Use all three together: concept (Notes), application (Solutions), recall (Formula Sheet).
Ques. How long should it take to finish all exercises of Class 12 Maths Chapter 7 Integrals?
Ans. A first pass through the 11 exercises of Class 12 Maths Chapter 7 Integrals typically takes 30-50 hours of focused study, including the time spent verifying each working against the Solutions PDF. The longer chapters in the Calculus block (Chapters 5, 6, 7, 8, 9) sit at the upper end of that range.
Ques. Are Class 12 Maths Chapter 7 Integrals NCERT Solutions useful for JEE Main?
Ans. Yes - JEE Main typically asks 3-4 questions from this chapter. The Expert's Solution tab on every back-exercise question shows the longer, JEE-style working with alternative methods sketched. For deeper competitive-exam practice, switch to the Class 12 Maths Chapter 7 Exemplar Solutions once the NCERT back-exercise feels comfortable.
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