CBSE Class 12 Integrals PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-3 MCQs/assertion-reason (1 mark), 3-4 short answers/indefinite integrals (2-3 marks), 1-2 definite integrals or properties-based (4-6 marks).
Repeated Topics: Standard integration formulas, substitution method, integration by parts, partial fractions, trigonometric integrals, definite integrals using properties, integration as limit of sum, integration of rational functions.
Difficulty: Medium to high; technique selection (substitution vs parts vs partial fractions) is crucial. More emphasis on definite integrals & properties in recent years.
Changes: Increased number of objective questions and application of properties to simplify definite integrals (2023–2025).

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Integrate ∫ dx / (x² + 6x + 13) or ∫ dx / √(x² - 4x - 5) using completing the square or standard form.	Short Answer (2-3 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Complete square → log or arc tan form.
Evaluate ∫ sinⁿx cosᵐx dx when one is odd (substitution) or both even (multiple angle identities).	Short Answer (3 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Save one sin/cos for du; rest use identities.
Evaluate definite integral ∫₀^π/2 sinⁿx dx or ∫₀^π/2 sin^m x cos^n x dx using properties (Wallis formula type).	Short/Long Answer (3-4 marks)	2022, 2023, 2024, 2025	Repeated 4x; Use I(n) reduction or property I = (n-1)/n × I(n-2).
Assertion: ∫ f(x) dx from a to b = - ∫ f(x) dx from b to a. Reason: Direction of integration reverses.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason correct.
Integrate by parts: ∫ x e^x dx or ∫ x² log x dx or ∫ x sin x dx.	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; ILATE rule: choose u & dv wisely; repeat if needed.
Integrate rational function: ∫ (2x+3)/(x² + 3x + 2) dx using partial fractions.	Short Answer (3 marks)	2022, 2023, 2024	Repeated 3x; Factor denominator → A/(x+1) + B/(x+2) → log form.
MCQ: ∫ dx / (x² + 2x + 2) = ? (a) log\|x²+2x+2\| (b) tan⁻¹(x+1) (c) (1/√2) tan⁻¹((x+1)/√2) etc.	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer involves arctan after completing square.
Using properties of definite integrals, evaluate ∫₀^π/2 log(sin x) dx or ∫₀^a f(x) dx where f(a-x) = f(x).	Long Answer (4-5 marks)	2023, 2024, 2025	Repeated 3x; Property I = ∫₀^a f(a-x) dx → useful for symmetric limits.
Case-based: Given integral with substitution hint or definite integral to evaluate using properties or by parts.	Case-Based (4-6 marks)	2023, 2025	Repeated 2x; Focus on technique identification.
Express ∫ dx / √(a² - x²) or ∫ dx / √(x² - a²) or ∫ dx / √(x² + a²) in standard form.	Short Answer (2 marks)	2021 Term 2, 2024	Repeated 2x; arcsin, log, arctanh forms respectively.

Best Format for Preparation and Answering

Study Format: Memorize all standard integrals (12–15 forms). Practice substitution, parts, partial fractions separately. Make a table of trig integrals & reduction formulas. Solve 15–20 PYQs daily.
Answer Format: Indefinite: +C always. Substitution: Show u = ..., du = .... Definite: Show limits change if substituted. Properties: Write property used → simplify step-by-step.
Tips: Master properties of definite integrals (very high repeat & save time). Practice identifying method quickly. Time: 12–15 mins for 4–6 mark questions. Scoring chapter if techniques are clear.

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