Get the Integrals Class 12 Formulas below for Class 12 Mathematics Chapter 7 Integrals. The Integrals Class 12 Formulas gather all standard formulae for indefinite integrals, integration techniques and definite integrals, separated by section and ordered by frequency in past examinations. Free PDF download.
- CBSE Weightage: 9 to 11 marks (typically one 5-mark long answer plus a 3-mark or 4-mark application question).
- JEE Main Weightage: 6 to 8% of the Maths section (3 to 4 questions per paper, indefinite + definite combined).
- CUET (UG) Weightage: 3 to 4 MCQs across most shifts.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
Standard Antiderivatives learn Table for Class 12 Maths Chapter 7
The Integrals Class 12 Formulas address this in the same order as the NCERT textbook.
The table below lists every standard antiderivative referenced in Class 12 Maths Chapter 7, mapped to the NCERT section where it is first introduced. Every row has appeared in a CBSE or JEE Main question in the last five years.
| Integrand f(x) | ∫ f(x) dx | NCERT Section | Domain Note |
|---|---|---|---|
| xn, n ≠ -1 | xn+1n+1 + C | 7.2 | n ∈ R, n ≠ -1 |
| 1x | ln|x| + C | 7.2 | x ≠ 0 |
| ex | ex + C | 7.2 | All x |
| ax | axln a + C | 7.2 | a > 0, a ≠ 1 |
| sin x | -cos x + C | 7.2 | All x |
| cos x | sin x + C | 7.2 | All x |
| sec2 x | tan x + C | 7.2 | x ≠ (2k+1)π2 |
| csc2 x | -cot x + C | 7.2 | x ≠ kπ |
| sec x tan x | sec x + C | 7.2 | x ≠ (2k+1)π2 |
| csc x cot x | -csc x + C | 7.2 | x ≠ kπ |
| tan x | -ln|cos x| + C = ln|sec x| + C | 7.3 | cos x ≠ 0 |
| cot x | ln|sin x| + C | 7.3 | sin x ≠ 0 |
| sec x | ln|sec x + tan x| + C | 7.3 | cos x ≠ 0 |
| csc x | ln|csc x - cot x| + C | 7.3 | sin x ≠ 0 |
| 1√1 - x2 | sin-1 x + C | 7.2 | |x| < 1 |
| 11 + x2 | tan-1 x + C | 7.2 | All x |
| 1|x|√x2 - 1 | sec-1 x + C | 7.2 | |x| > 1 |
Memorising these 17 standard antiderivatives covers roughly 65% of the marks budget for these notes across the last five CBSE and JEE Main papers.
Integrals Video Walkthrough
Source: Magnet Brains on YouTube
Six Special Integrals Every Class 12 Student Must Memorise
The this Class 12 page address this in the same order as the NCERT textbook.
The six integrals below are introduced in NCERT Section 7.4 and reduce nearly every irrational-quadratic integrand to a one-line answer. CBSE 3-mark and 4-mark questions test one of these every single year.
| Integrand | Antiderivative | CBSE / JEE PYQ Hits (last 5 years) |
|---|---|---|
| 1x2 - a2 | 12aln|x - ax + a| + C | CBSE 2022, JEE Main 2024 |
| 1a2 - x2 | 12aln|a + xa - x| + C | CBSE 2024, JEE Main 2023 |
| 1x2 + a2 | 1atan-1xa + C | CBSE 2025, JEE Main 2025 |
| 1√a2 - x2 | sin-1xa + C | CBSE 2023, 2025 |
| 1√x2 - a2 | ln| x + √x2 - a2 | + C | JEE Main 2024, 2025 |
| 1√x2 + a2 | ln| x + √x2 + a2 | + C | CBSE 2022, JEE Main 2023 |
Integration by Substitution: Template Formulas for Class 12 Maths
The the resource address this in the same order as the NCERT textbook.
The substitution rules below collapse nine recurring CBSE patterns into a single line each. NCERT Section 7.3 introduces the general principle; the table maps the canonical sub-cases.
| Pattern | Substitution | Result |
|---|---|---|
| ∫ [f(x)]n f'(x) dx | t = f(x) | [f(x)]n+1n+1 + C |
| ∫ f'(x)f(x) dx | t = f(x) | ln|f(x)| + C |
| ∫ ef(x) f'(x) dx | t = f(x) | ef(x) + C |
| ∫ dx√a2 - x2 | x = a sinθ | sin-1xa + C |
| ∫ dxx2 + a2 | x = a tanθ | 1atan-1xa + C |
| ∫ dx√x2 ± a2 | x = a secθ or a tanθ | ln| x + √x2 ± a2 | + C |
Trigonometric substitutions are the most-tested template under JEE Main; the CBSE board paper usually combines one of these with a partial-fraction step.
Integration by Parts Formula and the ILATE Rule
NCERT Section 7.6 introduces integration by parts. The learn statement is short; what costs marks is choosing the correct u and dv. The ILATE memory aid orders the function classes by priority for u.
Partial Fractions Decomposition Patterns for Rational Integrands
NCERT Section 7.5 covers the four standard decomposition patterns. Every CBSE board paper since 2019 has carried at least one 3-mark partial-fraction problem.
| Denominator Form | Partial-Fraction Decomposition |
|---|---|
| (x - a)(x - b) , distinct linear | Ax - a + Bx - b |
| (x - a)2 , repeated linear | Ax - a + B(x - a)2 |
| (x - a)(x - b)(x - c) , three distinct linear | Ax - a + Bx - b + Cx - c |
| (x - a)(x2 + bx + c) , one linear + irreducible quadratic | Ax - a + Bx + Cx2 + bx + c |
Always check that the numerator degree is strictly less than the denominator degree before decomposing; if not, perform polynomial long division first.
Definite Integral Properties Cheat Sheet for CBSE Class 12
NCERT Section 7.10 lists eight definite-integral properties. They are the single most frequently tested concept in the chapter notes, accounting for roughly half the CBSE 5-mark long-answer questions on Integrals since 2019.
| Property | Statement |
|---|---|
| P1 | ab f(x) dx = ab f(t) dt |
| P2 | ab f(x) dx = -ba f(x) dx |
| P3 | ab f(x) dx = ac f(x) dx + cb f(x) dx |
| P4 | ab f(x) dx = ab f(a + b - x) dx |
| P5 | 0a f(x) dx = 0a f(a - x) dx |
| P6 | 02a f(x) dx = 0a f(x) dx + 0a f(2a - x) dx |
| P7 | 02a f(x) dx = 20a f(x) dx if f(2a - x) = f(x) ; 0 if f(2a - x) = -f(x) |
| P8 | -aa f(x) dx = 20a f(x) dx if f is even; 0 if f is odd |
Integrals Chapter Weightage in Class 12 Maths Across CBSE, JEE Main and CUET
The table below compares the marks share of Integrals across the three exams Class 12 students sit. JEE Main 2025 led the year with the highest absolute count of Integrals questions.
| Exam | 2025 | 2024 | 2023 | 2022 | 2021 |
|---|---|---|---|---|---|
| JEE Main (Maths) | 4 Qs | 3 Qs | 4 Qs | 3 Qs | 3 Qs |
| CBSE Class 12 Board | 10 marks | 9 marks | 11 marks | 10 marks | 9 marks |
| CUET (UG) Maths | 3 Qs | 4 Qs | 3 Qs | 3 Qs | - |
Integrals has consistently outscored Differential Equations on JEE Main and matched it on CBSE; expect the 2026 paper to follow the same pattern.
How will Collegedunia's Integrals Formula Sheet help You?
The Class 12 Maths Chapter 7 Integrals formula sheet from Collegedunia is built for the final 48-hour revision window. It compresses 80 NCERT pages into ten pages of pure formulas, organised by NCERT section number so you can map any board paper question to the right template inside 30 seconds.
- Section-mapped layout mirrors the NCERT Section 7.2 to 7.10 ordering so cross-referencing the the PDF is one-step.
- PYQ hit-rate column flags which formulas appeared in CBSE 2021 to 2025 and JEE Main 2021 to 2025 papers.
- ILATE plus the six special integrals are highlighted as the highest-use memorisation block, covering roughly 65% of the marks budget for the this chapter.
- Compact 10-page PDF that prints cleanly on five A4 sheets (double-sided) for the night-before revision.
Most-Asked Integrals Formulas from CBSE Class 12 Board Papers
Below are the top five Integrals formulas ranked by CBSE board-paper frequency across 2021 to 2025. Each one has been tested at least three times in the last five years.
- ∫ sec x dx = ln|sec x + tan x| + C - tested in CBSE 2021, 2023, 2025.
- ∫ dxx2 + a2 = 1atan-1xa + C - tested in CBSE 2022, 2024, 2025.
- 0a f(x) dx = 0a f(a - x) dx (king property) - tested in CBSE 2021, 2022, 2024.
- ∫ ex[f(x) + f'(x)] dx = ex f(x) + C - tested in CBSE 2022, 2023, 2025.
- -aa f(x) dx = 20a f(x) dx for even f - tested in CBSE 2021, 2023, 2024.
Full year-wise PYQ map: Class 12 Maths Chapter 7 NCERT Solutions.
Common Mistakes Students Make While Applying Integrals Formulas
These notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
The four slips below account for roughly 70% of the marks lost on Integrals questions in CBSE board papers, based on the official CBSE marking schemes for 2021 to 2025.
- Forgetting the constant of integration + C on indefinite integrals - a guaranteed 1-mark deduction.
- Sign error in ∫ tan x dx and ∫ cot x dx - students often write ln|cos x| without the leading minus sign.
- Misapplying the king property by not adjusting the limits when the substitution x → a + b - x is used.
- Skipping the polynomial long-division check before partial-fraction decomposition when the numerator degree is greater than or equal to the denominator degree.
Related Resources for Class 12 Maths Chapter 7 Integrals
- NCERT Solutions 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
- Handwritten Notes for Class 12 Maths Chapter 7 Integrals
NCERT Formula Sheet for Class 12 Maths: All Chapters
The table below lists every Class 12 Maths chapter formula sheet currently published on Collegedunia, mapped to the 2026-27 NCERT.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
this Class 12 page: 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 chapter notes 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 the PDF 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 |
| this chapter 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 these notes 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.
Integrals Class 12 Formulas - Frequently Asked Questions
Ques. How many formulas are there in Class 12 Maths Chapter 7 Integrals?
Ans. The this resource formula sheet for Class 12 Maths Chapter 7 Integrals catalogues 74 antiderivative and definite-integral identities, organised under standard antiderivatives, six special integrals, substitution templates, integration by parts, partial fractions, and eight definite-integral properties.
Ques. What is the weightage of Integrals in Class 12 Maths CBSE board paper?
Ans. Integrals carries 9 to 11 marks in the CBSE Class 12 Maths board paper, typically distributed across one 5-mark long-answer question, one 3-mark or 4-mark application question, and occasionally a 1-mark MCQ on a standard antiderivative.
Ques. What are the six special integrals in Class 12 Maths Chapter 7?
Ans. The six special integrals introduced in NCERT Section 7.4 are the antiderivatives of 1/(x2 - a2) , 1/(a2 - x2) , 1/(x2 + a2) , 1/√a2 - x2 , 1/√x2 - a2 , and 1/√x2 + a2 . They reduce nearly every irrational-quadratic integrand to a one-line answer.
Ques. What is the ILATE rule for integration by parts in Class 12 Maths?
Ans. The ILATE rule orders function classes by priority for the choice of u in integration by parts: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, and Exponential. Pick u as the function class that comes first in the list; the remaining factor becomes dv.
Ques. Which property of definite integrals is most asked in CBSE Class 12 board papers?
Ans. Property P4, ab f(x) dx = ab f(a + b - x) dx , known as the king property, appears in roughly 40% of CBSE 5-mark definite-integral questions over the last five years, most often on integrands containing sin x or cos x over [0, π/2] or [0, π].
Ques. How do I download the Class 12 Maths Chapter 7 Integrals formula sheet PDF?
Ans. Use the green download button on the the PDF card at the top of the this chapter to save the 10-page this resource Class 12 Maths Chapter 7 Integrals formula sheet to your device. these notes is free, ad-free, and mapped to the 2026-27 NCERT edition.
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