Class 12 Maths Integrals: Important Questions, Formula Sheet & Board Exam Tips (2026-27)

Class 12 Maths Integrals: Important Questions, Formula Sheet & Board Exam Tips
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If you’ve already gone through your class 12 mathematics textbook exercises and are now staring down board exams wondering “what actually gets asked from Integrals,” you’re not alone – this is the chapter where most Class 12 students lose easy marks, not because the concepts are impossible, but because there’s no clear map of which formulas repeat every year and which question types examiners keep coming back to. This guide skips the textbook walkthrough and goes straight to what matters for CBSE 2026-27: a compact formula sheet you can revise in one sitting, the question patterns that show up almost every year in board papers and the small errors that quietly cost students 1-2 marks per question. Think of it as the revision sheet you’d want the night before your exam.

Master Integrals for Class 12 Boards 2026-27: Formula Sheet + Most Repeated Questions

Integration is one of those chapters where the syllabus looks long, but the exam pattern is surprisingly predictable once you’ve seen a few years of question papers side by side. CBSE tends to draw from a fairly consistent pool of question types year after year – which means smart revision beats blind practice here. This guide is built specifically for that: fast recall, pattern recognition and the mistakes that trip up even well-prepared students.

Why Integrals Deserves Extra Attention This Year

For the 2026-27 CBSE session, Integrals continues to carry significant weight in the board paper – typically contributing anywhere from 8 to 11 marks across short and long-answer questions, and it’s also foundational for the next chapter, Application of Integrals, which asks you to actually use these formulas to find areas under curves. Skip a solid grip on this chapter and Chapter 8 becomes twice as hard.

It’s also one of the most tested chapters in JEE Main and NEET, so time spent here compounds – this isn’t a “learn it and forget it” chapter.

Quick-Recall Formula Sheet

Keep this table open on a second tab while solving practice questions – don’t try to memorize everything in one go, use it as a reference until it sticks.

FunctionIntegral
xโฟxโฟโบยน/(n+1) + C; n โ‰  -1
1/xlog|x| + C
eหฃeหฃ + C
aหฃaหฃ/log a + C; a > 0, a โ‰  1
sin x-cos x + C
cos xsin x + C
secยฒxtan x + C
cosecยฒx-cot x + C
sec xยทtan xsec x + C
cosec xยทcot x-cosec x + C
1/โˆš(1-xยฒ)sinโปยนx + C
1/(1+xยฒ)tanโปยนx + C
1/(xโˆš(xยฒ-1))secโปยนx + C

Standard substitutions worth memorizing:

  • โˆš(aยฒ โˆ’ xยฒ) โ†’ put x = a sin ฮธ
  • โˆš(aยฒ + xยฒ) โ†’ put x = a tan ฮธ
  • โˆš(xยฒ โˆ’ aยฒ) โ†’ put x = a sec ฮธ

Integration by parts (ILATE rule) – priority order for choosing the first function:

  1. Inverse trigonometric
  2. Logarithmic
  3. Algebraic
  4. Trigonometric
  5. Exponential




Question Types That Repeat Almost Every Year

Rather than guessing what to practice, here’s what actually shows up based on recurring board patterns:

  1. Integration using standard substitution – expect at least one 2-mark question testing a direct substitution trick
  2. Integration by parts – almost guaranteed every year, usually a 3-mark question, frequently involving log or inverse trig functions
  3. Integration using partial fractions – a reliable 3-5 mark question, especially with rational functions where the denominator factors nicely
  4. Definite integrals using properties – this is the topic students underestimate the most; properties like P4 and P6 (King’s Property) show up disguised in ways that look unfamiliar at first glance
  5. Miscellaneous exercise-style mixed questions – combining two techniques in one problem, usually the highest-mark question from this chapter (5 marks)

Common Mistakes That Cost Marks

These aren’t conceptual gaps – they’re small habits that quietly reduce scores, even for students who understand the topic well.

  • Forgetting the “+C” – sounds trivial, but examiners do deduct marks for a missing constant of integration
  • Sign errors in substitution – especially when substituting trigonometric identities; always double-check the derivative of your substitution before proceeding
  • Misapplying ILATE – picking the wrong “first function” in integration by parts leads to a longer, messier solution or a dead end
  • Skipping the check for definite integral limits – swapping limits without adjusting the sign is a very common slip
  • Not simplifying before integrating – many questions become far easier after a quick algebraic simplification (splitting fractions, using trig identities) that students skip in a hurry




A Practical 5-Day Revision Plan

Day FocusGoal
Day 1Formula sheet + basic indefinite integralsBuild recall speed
Day 2Integration by substitution & by partsPattern recognition
Day 3Partial fractionsPractice factoring + splitting
Day 4Definite integrals + propertiesMaster P1โ€“P6 properties
Day 5Previous year mixed questionsTimed practice, simulate exam conditions

Types of Questions to Expect by Marks Weightage

  • 1 Mark (MCQ/Assertion-Reason): Direct formula-based, standard integral identification
  • 2 Marks: Simple substitution or direct application of a formula
  • 3 Marks: Integration by parts or partial fractions
  • 5 Marks: Mixed technique problems, often combining substitution with by-parts, or a longer definite integral using multiple properties

Where This Fits in Your Revision

This page is meant to be your quick-reference companion, not a replacement for working through the actual textbook exercises step by step. Once you’ve used the formula sheet and question patterns here to sharpen your recall, the natural next step is practicing the complete NCERT exercise solutions – walking through 7.1 to 7.10 and the Miscellaneous Exercise with full step-by-step working, which is exactly what you’ll find in the detailed NCERT Solutions for Class 12 Maths Chapter 7 -Integrals on Tiwari Academy, updated for the 2026-27 CBSE session in both English and Hindi medium.




Frequently Asked Questions

How many marks does Integrals carry in the Class 12 Maths board exam 2026-27?

Integrals typically contributes around 8-11 marks in the CBSE board paper, spread across MCQs, short-answer and long-answer questions. It’s also directly linked to Chapter 8 (Application of Integrals), so a strong grip here indirectly supports marks in that chapter too.

What is the fastest way to revise Integrals just before the exam?

Focus on the formula sheet first for quick recall, then run through 8-10 previous year questions covering integration by parts, partial fractions and definite integral properties – these three areas account for most of the marks in this chapter.

Which topic in Integrals is most commonly asked in board exams?

Integration by parts and definite integrals using properties (especially King’s Property) are the most consistently repeated question types across recent CBSE papers, usually worth 3-5 marks each.

Is Integrals important for JEE and NEET as well as boards?

Yes. Integrals is one of the most heavily weighted calculus topics in both JEE Main and NEET, so time invested in mastering this chapter for boards directly benefits competitive exam preparation too.

What’s the most common mistake students make in Integrals questions?

Forgetting the constant of integration (+C) and sign errors during substitution are the two most frequent, avoidable mistakes – both are simple habits to fix with careful practice rather than requiring deeper conceptual work.

Should I memorize all integration formulas or understand the derivation?

A mix of both works best. Memorize the standard formulas (from the table above) for speed during the exam, but understand at least the derivation of 2-3 key ones (like integration by parts) so you can apply the logic to unfamiliar variations examiners sometimes introduce.