JEE Main Maths · Integration PYQ
JEE Main Integration Previous Year Questions (2002–2025)
Integration is the single highest-weightage Mathematics chapter in JEE Main — 8% across 10 years with 2 questions per session routinely. Along with Differentiation and Limits, it forms the Calculus core that drives ~25% of the Maths section. A strong Integration performance is nearly non-negotiable for a 75+ Maths score.
Integration PYQs from 2002 to 2025, split across definite integration, indefinite techniques, by-parts, partial fractions and property-based evaluations. Every solution is KaTeX-rendered with the technique used tagged upfront, so you build a personal shortcut library.
Integration at a Glance
Key Sub-Topics & What's Tested
Standard Integration Formulas
List of ~30 standard results (polynomial, trigonometric, exponential, logarithmic, inverse trig). PYQs assume instant recall of all 30.
Integration by Substitution
u-substitution strategy, trigonometric substitutions, rationalising substitutions for root-of-quadratic expressions.
Integration by Parts
ILATE rule, repeated by-parts, DI method for shortcut, by-parts for inverse trigonometric and logarithmic integrands.
Partial Fractions
Rational function decomposition, cases for linear/quadratic denominators, combining with standard results.
Definite Integral Properties
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx, king's property for symmetric integrands, periodic integrand reduction.
Reduction Formulas
∫ sin^n x dx, ∫ cos^n x dx, ∫ tan^n x dx reduction patterns for definite integrals over standard intervals.
Area Under Curves
Area between curve and axis, between two curves, bounded regions, parametric & polar area integrals (rarely in Main).
Special Integrals
Integrals involving e^(ax)·sin(bx) or e^(ax)·cos(bx), integration of rational functions with irrational expressions.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Direct Formula Application | 25% | Evaluate ∫ x/(1+x²) dx using standard log result. |
| Substitution-Based | 30% | Choose the right u-substitution for ∫ x √(x²+1) dx. |
| Integration by Parts | 15% | Evaluate ∫ x · ln(x) dx using ILATE rule. |
| Definite Integral with Property | 20% | Evaluate ∫₀^(π/2) sin²x / (sin²x + cos²x) dx using king's property. |
| Area Bounded by Curves | 10% | Find area bounded by y = x² and y = x between x = 0 and x = 1. |
How to Solve Integration PYQs
- 1Lock the 30 standard integrals. These are the 30 results that show up across the vast majority of Integration PYQs. Without instant recall, you lose 90 seconds per question.
- 2Always try substitution first. Most Main-level integrals yield to a clever u-substitution. By-parts and partial fractions come later in the decision tree.
- 3Use ILATE for by-parts order. Inverse, Log, Algebraic, Trig, Exponential. Pick u as the first in ILATE that appears in the integrand.
- 4Learn the king's property. ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx collapses half of all definite integral PYQs to one-line algebra.
- 5Partial fractions for rational functions. If denominator factors cleanly into linear / quadratic terms, partial fractions. If not, try completing the square or clever substitution.
- 6Area PYQs: sketch first. Always draw the region before setting up the integral — the limits of integration follow from the sketch, not from algebra alone.
Common Mistakes That Cost Marks
- Missing the constant of integration in indefinite. + C is part of the answer. PYQs sometimes test whether you know which form of the constant is correct.
- Wrong sign in log-based integrals. ∫ 1/x dx = ln|x| + C (absolute value). Forgetting absolute value in sign-sensitive problems is a routine error.
- Forgetting chain rule in substitution. dx must be replaced with (du / u'(x)), not just du. This is where half of substitution errors come from.
- Wrong ILATE order for by-parts. Using x as u with cos(x) as dv works; reversing wastes 5 minutes.
- Not checking symmetry before long calculations. Many definite integrals reduce to 0 or symmetric halves — check for even/odd symmetry or king's property before integrating.
Related JEE Main Practice
Frequently asked questions
How many Integration PYQs should I solve?
Target 100+ Integration PYQs across 2010–2025. Integration has the highest Maths weightage and the highest template-repeat rate — you'll see same substitution patterns reused across years.
Are Integration PYQs harder in JEE Main or JEE Advanced?
Advanced. Main tests standard techniques (substitution, by-parts, partial fractions, properties). Advanced layers in non-standard substitutions, Walls limit tricks, and integrals that require multi-step clever algebra first. Main-level PYQ mastery is the foundation for Advanced.
Do I need to memorise reduction formulas?
For JEE Main, memorise the three most-used: ∫ sin^n x dx, ∫ cos^n x dx, ∫ tan^n x dx between standard limits. Full reduction formula tables are Advanced-level overkill for Main.
What's the most-tested king's property PYQ template?
∫₀^(π/2) sin^n x / (sin^n x + cos^n x) dx = π/4 for any n. This appears verbatim or as a close variant in ~8 PYQs across 2010–2025. Free marks if recognised.
How are Area Under Curves PYQs structured?
Usually: sketch region, identify limits of integration, integrate. The sketching step is critical. Common templates: area between parabola and line, between two parabolas, bounded by trigonometric functions.
Does Integration connect with Differential Equations in PYQs?
Yes — first-order DEs often reduce to evaluating an integral. Variable separable and linear DEs both end with a required integration. Strong Integration fluency pays off in DE PYQs too.
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