India's #1 AI Tutorimportant-questions · Mathematics · Chapter 7

Important Questions: CBSE Class 12 Mathematics Chapter 7 Integrals

Integrals form the backbone of calculus in CBSE Class 12 Mathematics, bridging algebraic techniques with geometric applications. This chapter demands both procedural fluency and conceptual clarity. Students in coaching hubs across Kota, Delhi's Mukherjee Nagar, and Chennai's T Nagar spend considerable time mastering the diverse integration methods. This collection mirrors actual board exam patterns from 2020-2025, offering targeted practice across all mark categories with detailed solutions.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓Chapter 7 Integrals typically accounts for 15-18 marks in the CBSE Class 12 Mathematics board exam, distributed across various question types.
✓Integration by substitution and integration by parts together form the basis for 40-50% of questions from this chapter.
✓Definite integrals questions often test the fundamental theorem of calculus and properties like symmetry, contributing 6-8 marks annually.
✓Case-based questions introduced from 2020-21 onwards frequently use real-life contexts requiring definite integration for area or accumulation problems.
✓Common errors include forgetting the constant of integration in indefinite integrals and incorrect application of limits in definite integrals.
✓CBSE examiners reward clear stepwise working—showing substitution steps, applying formulae explicitly, and simplifying systematically earns full marks even if the final answer has minor errors.
✓Practising previous year papers reveals that questions on rational function integration using partial fractions appear almost every year as 3-mark or 5-mark problems.

Chapter Overview and Marks Weightage in CBSE Class 12 Board Exam

Chapter 7 Integrals is the second-highest weighted chapter in CBSE Class 12 Mathematics after Calculus (Differentiation). The 2024-25 examination pattern allocates approximately 15-18 marks to this chapter across the 80-mark theory paper. The chapter builds on NCERT Class 12 Mathematics Part I textbook content, covering indefinite integrals through various methods and definite integrals with the fundamental theorem of calculus. Questions appear as 1-mark objective type (MCQs), 2-mark very short answer questions, 3-mark short answer questions, and 5-mark long answer or case-based questions. The CBSE marking scheme rewards methodical working—students who show substitution clearly, state formulae before applying them, and simplify step-by-step typically secure 90-95% of available marks even with calculation slips. The 2023 Delhi region topper scored full marks in Integrals by practising diverse question types rather than memorising solutions. Understanding the weightage helps students allocate roughly 25-30 hours of focused practice to this chapter during board preparation, ensuring mastery of substitution, parts, partial fractions, and definite integral properties that recur yearly in CBSE papers.

Indefinite integrals (substitution, trigonometric, standard forms): 6-8 marks
Integration by parts and partial fractions: 4-6 marks
Definite integrals and fundamental theorem: 5-7 marks
Case-based or application problems: 4-5 marks
Typical distribution: one 5-mark question, two 3-mark questions, one 2-mark question, and 1-2 MCQs

1-Mark Questions: Multiple Choice and Very Short Answer

One-mark questions in CBSE Class 12 Mathematics Chapter 7 test direct formula recall, standard integral recognition, and quick conceptual checks. These MCQs or fill-in-the-blank questions typically appear in Section A of the board paper. Students should aim to solve each in under 60 seconds during the exam. Common themes include identifying the correct substitution, recognising standard forms from the NCERT integration table, evaluating simple definite integrals using properties, and selecting the correct antiderivative from options. CBSE often uses these questions to test common misconceptions—for instance, offering answer choices that differ only by the constant of integration or sign. Practising 50-60 such questions builds speed and accuracy. In the 2023 paper, one MCQ asked students to identify ∫(1/√(1 - x²)) dx from four options; students who had memorised the standard inverse trigonometric integrals scored immediately. Another asked the value of ∫₀^(π/2) sin x dx, testing the fundamental theorem with basic limits. Smart students tackle these first in the exam to secure 4-5 quick marks and build confidence before moving to longer questions.

2-Mark Questions: Short Working and Direct Methods

Two-mark questions in Class 12 Mathematics Chapter 7 Integrals demand concise working within 3-4 lines, testing a single integration technique or a straightforward definite integral. These questions appear in Section B and typically involve one substitution step, one standard formula application, or evaluation of a simple definite integral using limits. CBSE marking schemes allocate 1 mark for correct method and 1 mark for the final answer. Students must show the substitution or formula used—writing the answer directly without working rarely earns full marks. Common question types include integrating functions like ∫(ax + b)ⁿ dx using substitution, finding ∫tan x dx or ∫sec x dx using standard results, evaluating definite integrals of polynomial or trigonometric functions over symmetric intervals using properties, and anti-differentiating exponential or logarithmic functions. In the 2022 CBSE paper, a 2-mark question asked students to evaluate ∫(cos 2x)/(cos x + sin x)² dx—requiring the substitution t = cos x + sin x and dt = (-sin x + cos x)dx after rewriting the numerator. Students who identified the pattern scored easily; those who attempted direct integration struggled. Practising 20-25 such problems builds pattern recognition essential for speed in the board exam.

3-Mark Questions: Intermediate Techniques and Multi-Step Problems

Three-mark questions form the core of CBSE Class 12 Mathematics Chapter 7 Integrals, testing integration by parts, partial fractions, substitution with trigonometric identities, and definite integrals requiring property application. These questions appear in Section C and demand 6-8 lines of systematic working. CBSE examiners allocate marks for method selection (0.5-1 mark), correct intermediate steps (1-1.5 marks), and final simplified answer (1 mark). Students should label each step—writing 'Using integration by parts with u = x, dv = eˣdx' earns clarity marks. Common themes include integrating rational functions by partial fraction decomposition (splitting (3x+5)/((x-1)(x+2)) into A/(x-1) + B/(x+2)), applying integration by parts for products like ∫x·sin x dx or ∫x²·eˣ dx, using trigonometric substitutions for integrals involving √(a²-x²) or √(x²+a²), and evaluating definite integrals using properties of even/odd functions or symmetry. The 2024 CBSE paper included ∫₀^(π/2) (x·sin x)/(1 + cos²x) dx, requiring the property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx to simplify before integration. Students who knew this property solved it in 4 lines; others spent 10 minutes without success. Mastering 30-35 such problems across all techniques is essential for scoring 27+ out of 30 in the Integrals section.

5-Mark Questions: Integration by Parts, Partial Fractions, and Case-Based Problems

Five-mark questions in CBSE Class 12 Mathematics Chapter 7 Integrals are the most demanding, often combining multiple techniques or presenting real-world case studies requiring definite integration. These appear in Section D or E and expect 12-15 lines of detailed working with clear justification at each step. CBSE marking schemes distribute marks across method identification (1 mark), correct application of technique (2 marks), accurate computation (1.5 marks), and final answer (0.5 mark). Typical question types include integrating rational functions where the degree of numerator equals or exceeds the denominator (requiring polynomial division first), successive application of integration by parts (e.g., ∫x²·sin x dx needs parts applied twice), definite integrals evaluated using advanced properties or substitution that transforms limits, and case-based questions where students read a 4-5 line scenario—such as water flow rate or population growth—and set up then solve a definite integral. The 2023 CBSE paper presented a case study about a company's marginal profit function P'(x) = 3x² - 6x + 5, asking students to find total profit from x=0 to x=10 units by integrating, applying the fundamental theorem, and interpreting the result economically. Students who structured their answer with clear substitution of limits and units in the final statement scored all 5 marks. Practising 15-20 such comprehensive problems, including at least 5 case-based scenarios, prepares students for the rigour CBSE expects at this level.

Always perform polynomial long division when the degree of numerator ≥ degree of denominator before attempting partial fractions
For repeated integration by parts, tabulate u, dv, du, v in columns to avoid sign errors
In case-based questions, underline the given function and what is being asked before setting up the integral
Show limit substitution separately: write F(b) - F(a) with actual values, then compute—examiners award marks for clarity

Case-Based and Application Questions from Integrals

Since the 2020-21 session, CBSE introduced case-based questions worth 4-5 marks that present a real-life context followed by sub-questions. For Chapter 7 Integrals, these scenarios typically involve rate functions requiring integration to find total quantity—such as velocity to displacement, marginal cost to total cost, or population growth rate to total population over an interval. The case study includes a paragraph (3-4 sentences) describing the situation, a graph or equation representing the rate, and 2-3 sub-parts asking students to set up and evaluate definite integrals, interpret the result, or compare outcomes under different conditions. CBSE mark allocation is usually 1+2+2 marks across the sub-parts. Students must extract the integrand from the context, identify correct limits from the problem statement, apply the fundamental theorem of calculus, and write the answer with appropriate units or interpretation. In the 2024 board exam, one case study described a tank filling at a rate R(t) = 3t² + 2t litres per minute for 0 ≤ t ≤ 5 minutes. Sub-part (i) asked for total water filled (requiring ∫₀⁵(3t²+2t)dt), sub-part (ii) asked at what time the tank held exactly 50 litres (setting the integral from 0 to t equal to 50 and solving), and sub-part (iii) tested understanding of average rate. Students from Delhi's Rajendra Nagar coaching centres who had practised similar NCERT exemplar problems scored consistently well. The key is reading carefully—underlining the rate function, the limits, and what quantity is being accumulated—before writing the integral. Practising 8-10 diverse case studies from previous years and sample papers is essential preparation.

Identify whether the given function represents a rate (derivative) or the quantity itself—only rates need integration
Convert word problems into mathematical notation: 'from start to 10 seconds' becomes limits 0 to 10
Always include units in your final answer for application problems (litres, rupees, metres, etc.)
If asked for average value, use the formula (1/(b-a))∫ₐᵇ f(x)dx after evaluating the integral

How CBSE Frames Questions from Chapter 7 Integrals

CBSE question paper setters follow a predictable pattern when designing Integrals questions, informed by the learning outcomes specified in the NCERT Class 12 Mathematics curriculum and refined through annual question paper analysis meetings held at regional centers in Delhi, Chennai, and Ajmer. Typically, the blueprint ensures coverage of all major methods: at least one question each on substitution, integration by parts, and partial fractions, plus one or two on definite integrals and properties. The difficulty progression is deliberate—1-mark questions test standard forms from the NCERT integration table verbatim, 2-mark questions require one substitution or direct formula, 3-mark questions combine two techniques or involve simplification before integration, and 5-mark questions demand multi-step reasoning or real-world modelling. Examiners often present integrals where the 'obvious' method fails, nudging students toward algebraic manipulation first—for instance, adding and subtracting terms to create a known derivative in the numerator for rational functions, or using trigonometric identities to convert products into sums. The 2025 sample paper included ∫(sin x + cos x)/√(1 + sin 2x) dx, which simplifies only after recognising that 1 + sin 2x = (sin x + cos x)². CBSE also favours questions where different methods yield the same answer, testing conceptual flexibility—integration by parts and substitution for ∫x·e²ˣ dx both work. Understanding this pattern helps students allocate practice time: spend 40% of time on technique mastery, 30% on mixed problems requiring method selection, 20% on definite integrals and properties, and 10% on case studies. Coaching institutes in Kota and Delhi analyse 10 years of papers to identify recurring question frames, but self-study students can achieve the same insight by solving the last 5 years' papers with a focus on question structure rather than just answers.

Substitution questions often hide the substitution—look for a function and its derivative present simultaneously in the integrand
Integration by parts questions typically involve products of polynomial, exponential, trigonometric, or logarithmic functions; use the ILATE rule to choose u
Partial fractions appear when the denominator is factorable into distinct or repeated linear factors, or irreducible quadratics
Definite integral properties (even/odd functions, ∫₀ᵃf(x)dx = ∫₀ᵃf(a-x)dx) are tested via problems that are difficult to integrate directly but simplify elegantly using properties
Case-based questions always provide all necessary information within the passage—do not assume external knowledge

Common Mistakes Students Make in Integrals Questions

CBSE Class 12 board exam answer scripts reveal recurring errors in Chapter 7 Integrals that cost students 20-30% of available marks. The most frequent mistake is omitting the constant of integration C in indefinite integrals—even though the question may carry 3 marks, examiners deduct 0.5-1 mark for this oversight because it indicates incomplete understanding of the antiderivative concept. Another common error is incorrect substitution of limits in definite integrals: students evaluate F(b) - F(a) but use the wrong sign or swap the limits, especially after a u-substitution where the new limits must be recalculated. In integration by parts, sign errors proliferate when students fail to carefully track the negative sign in ∫u dv = uv - ∫v du; writing out each term separately prevents this. Partial fraction decomposition trips students when the denominator has repeated factors or irreducible quadratics—many forget that repeated factor (x-a)² requires both A/(x-a) and B/(x-a)² in the decomposition. In trigonometric integrals, students often misapply identities, such as writing sin²x = 1 - cos²x when (1 - cos2x)/2 is needed for integration. CBSE examiners in their 2024 post-exam report noted that in questions involving ∫₀^(π) f(x)dx where f is even or odd, many students computed the integral over the full range rather than using the property to simplify. Finally, in case-based questions, students lose marks by not stating units or misinterpreting what the integral represents—integrating a velocity function gives displacement, not distance travelled if the velocity changes sign. Coaching teachers in Hyderabad's Ameerpet and Pune's Kothrud emphasise error logs: students should maintain a notebook of mistakes from practice tests, review it weekly, and consciously avoid those patterns in subsequent problems. CBSETUTOR.ai offers targeted mistake analysis through its AI tutor, which flags common errors in uploaded homework photos and provides instant corrections—helping students self-correct before the board exam. At ₹999/month for unlimited access across all Class 6-12 subjects, the platform is increasingly popular among students in Tier-2 cities lacking quality coaching.

Always write + C at the end of indefinite integrals; make it a reflex action after writing the final expression
When substituting u = g(x), remember to change the limits: if x = a then u = g(a), if x = b then u = g(b)
In integration by parts, double-check the sign before the second integral term—it is always negative
For partial fractions with repeated roots, include all powers from 1 up to the multiplicity: (x-2)³ requires A/(x-2) + B/(x-2)² + C/(x-2)³
Verify definite integral answers by checking if the area interpretation makes sense (positive area for positive functions, zero area for odd functions over symmetric intervals)
In exam pressure, avoid skipping steps—write du =..., dv =... explicitly; examiners award marks for visible working

Strategy for Scoring Full Marks in Integrals Section

Maximising marks in CBSE Class 12 Mathematics Chapter 7 Integrals requires a blend of technique mastery, strategic time management during the exam, and presentation discipline. Start preparation by thoroughly solving all NCERT textbook exercises 7.1 through 7.11, ensuring you can reproduce every solution without peeking—these exercises form the template for 60-70% of board exam questions. Next, categorise integration techniques: create separate practice sets for substitution (20 problems), integration by parts (15 problems), partial fractions (15 problems), trigonometric integrals (10 problems), and definite integrals with properties (15 problems). Use CBSE sample papers and the last 5 years' board papers to identify which techniques appear most frequently and weight your practice accordingly. During the exam, attempt Integrals questions in the middle of your solving sequence—after you have secured marks in easier chapters like Matrices or Relations but before fatigue sets in. Allocate time proportionally: 1 mark per minute for MCQs, 3 minutes for 2-mark questions, 5-6 minutes for 3-mark questions, and 10-12 minutes for 5-mark questions. If you get stuck on a substitution, move on and return later rather than losing 5 minutes; other questions might jog your memory. In your answer script, underline or box final answers, write substitutions and formula names in the margin for easy examiner navigation, and leave one line blank between major steps so the script looks organised—examiners award grace marks more readily to neat, legible work. For definite integrals, always write the limits on the integral sign (∫ₐᵇ) rather than remembering to apply them later; this visual cue prevents the common error of forgetting limits. Finally, reserve 3-4 minutes at the end to recheck all indefinite integrals for the +C and verify limit substitutions in definite integrals. Students following this method in the 2024 board exams reported scoring 16-18 out of 18 in the Integrals section, with time left for other chapters. Personalised strategy refinement is available through platforms like CBSETUTOR.ai, where the AI tutor observes your solving patterns via photo uploads, identifies whether you are stronger in algebraic or trigonometric methods, and tailors practice sets to shore up weaker areas—all accessible 24×7 for just ₹999/month with a 3-day free trial.

Solve NCERT back exercises first, then proceed to exemplar problems, then previous years' papers in that sequence
Create a formula sheet with standard integrals (∫eˣdx, ∫(1/(1+x²))dx, etc.) and properties—revise it daily in the week before the exam
In the exam hall, read each Integrals question fully before starting; sometimes the last line hints at the method ('…using integration by parts')
Show all substitution steps: write 'Let u =...', 'du =...', 'dx =...'; examiners give step marks even if you do not reach the final answer
If the integral looks complex, try simplifying the integrand first—split fractions, use trigonometric identities, or factor—before choosing a method
Review your answers: recalculate one definite integral end-to-end if time permits, as limit substitution errors are easy to fix and recover full marks

Additional Practice Questions with Solutions

Sustained practice across varied problem types cements understanding and builds the speed necessary for CBSE board exam success. The following additional questions span all marks categories and mirror recent exam trends, including tricky substitutions, repeated integration by parts, and property-based definite integrals. Work through each question without looking at the solution first—time yourself to simulate exam pressure. After solving, compare your method and final answer with the model solution provided. Note any technique you found difficult and add similar problems to your practice queue. Successful Class 12 students in Jaipur's Vaishali Nagar and Lucknow's Gomti Nagar coaching centres solve at least 100 integration problems beyond NCERT during their board preparation, ensuring they encounter every possible question frame. These questions also illustrate how CBSE varies the same core concept across different function types—recognising the underlying pattern (e.g., ∫f'(x)/f(x)dx = ln|f(x)| + C) allows you to solve an entire class of problems instantly. Regular timed practice using these questions, ideally in 45-minute mock slots, builds both confidence and stamina for the 3-hour board paper marathon.

Frequently asked questions

How many marks does Chapter 7 Integrals carry in the CBSE Class 12 Mathematics board exam?+

Chapter 7 Integrals typically accounts for 15-18 marks out of 80 in the CBSE Class 12 Mathematics theory paper, making it one of the highest-weightage chapters. Questions appear as 1-mark MCQs, 2-mark, 3-mark, and 5-mark problems, plus case-based questions worth 4-5 marks.

What are the most important topics in Integrals for CBSE Class 12 boards?+

The most important topics are integration by substitution (including trigonometric substitutions), integration by parts using the ILATE rule, partial fraction decomposition for rational functions, definite integrals with the fundamental theorem of calculus, and properties of definite integrals (even/odd functions, ∫₀ᵃf(x)dx = ∫₀ᵃf(a-x)dx). Together these cover 90% of exam questions.

How should I prepare for case-based questions in Integrals?+

Case-based questions present a real-world scenario requiring definite integration. Practise extracting the rate function, identifying limits from context, setting up the integral, and interpreting the result with units. Solve 8-10 case studies from CBSE sample papers, focusing on contexts like marginal cost-to-total cost, velocity-to-displacement, and population growth over time.

What is the ILATE rule for integration by parts?+

ILATE is a mnemonic for choosing which function to differentiate (u) when applying integration by parts ∫u dv = uv - ∫v du. Preference order: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Pick u as the function appearing first in ILATE order; the other becomes dv. For example, in ∫x·sin x dx, x (algebraic) comes before sin x (trigonometric), so u = x.

Why do I keep forgetting the constant C in indefinite integrals?+

Omitting +C is a reflex error under exam pressure. Prevent it by making C the last character you write for every indefinite integral—treat it as part of the final answer symbol, not an afterthought. Practise writing answers as 'expression + C' in one breath. CBSE deducts 0.5-1 mark per omission, so this habit is worth 2-3 marks across the paper.

How do I know when to use partial fractions versus substitution?+

Use partial fractions when the integrand is a rational function (polynomial over polynomial) and the denominator factors into linear or irreducible quadratic terms. Use substitution when you spot a function and its derivative both present, or when a substitution simplifies the integrand into a standard form. If unsure, try spotting a derivative first (substitution) before attempting algebraic decomposition (partial fractions).

What are the most common errors in definite integrals?+

Common errors include forgetting to change limits after u-substitution (if u = g(x), new limits are g(a) and g(b)), applying limits before completing the integration, sign errors in F(b) - F(a), and not simplifying the antiderivative before substituting limits. Always write [F(x)]ₐᵇ notation and show F(b) and F(a) separately before subtracting.

How many practice problems should I solve to master Integrals?+

Solve all NCERT textbook exercises (approximately 60 problems), all 15 miscellaneous exercise questions, 30-40 problems from NCERT Exemplar, and at least 50 problems from the last 5 years' CBSE papers and sample papers. In total, 150-160 well-chosen problems covering all techniques ensure thorough preparation and 16+ marks in this chapter.

Are previous years' CBSE Integrals questions repeated?+

Exact questions are rarely repeated, but question frames and techniques recur annually. For instance, every year features at least one rational function partial fraction question, one integration by parts with a polynomial and trigonometric product, and one definite integral using the symmetry property. Practising 2018-2024 papers reveals these patterns clearly.

How can CBSETUTOR.ai help me with Integrals preparation?+

CBSETUTOR.ai provides a 24×7 AI tutor accessible at ₹999/month (all subjects, classes 6-12). Upload a photo of any Integrals problem, and the AI delivers step-by-step solutions with method explanations. It identifies your weak areas—say, integration by parts—and generates targeted practice sets. The 3-day free trial lets you test it with Chapter 7 problems before committing, making it cost-effective for students without access to coaching in smaller cities.

Related resources

CBSE Class 12 Mathematics Chapter 7 Integrals Worksheet with Answers Class 12 Mathematics Chapter 7 Integrals — Formulas & Key Points Important Questions: CBSE Class 12 Mathematics Chapter 6 Application of Derivatives Class 12 Mathematics Chapter 6 Application of Derivatives — Formulas & Key Points AI Tutor for Class 12: The Smart Alternative to Tuition AI Tutor for Class 12 Accountancy: Learn Faster with Instant Help Class 9 Coordinate Geometry Chapter 3: Cartesian System & Plotting Points Important Questions: CBSE Class 9 Mathematics Chapter 2 Polynomials

Keep learning — related guides

Class 12Mathematics

Class 12 Mathematics Chapter 13 Probability — Formulas & Key Points

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 13 Probability Worksheet with Answers

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 13 Probability

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 12 Linear Programming

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 12 Linear Programming Worksheet with Answers

Class 12Mathematics

Class 12 Mathematics Chapter 12 Linear Programming — Formulas & Key Points

Ready to give your Class 12 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 12 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →