If you want to know how to study Definite Integration for JEE — Main or Advanced — this guide is built entirely from real previous-year papers. We pulled every question this chapter produced in the most recent cycles — 97 questions across JEE Main 2024, 2025 and 2026, plus 17 from JEE Advanced 2020–2026 — and sorted them by technique. What follows is the real pattern behind one of the most consistently heavy Calculus chapters on both exams.
Definite Integration carries a 6.53% weightage in JEE Main 2026, up 34.92% from 2025 — a significant jump that makes it one of the fastest-growing chapters we've tracked in this series. On JEE Advanced it sits at 2.94%, down 6.07% — smaller in raw share, but the chapter has appeared reliably across the last several Advanced papers. Between the two exams, this chapter's real story is Main: it's climbing fast, and a big share of that growth is concentrated in one specific question style.
| Year | Questions (MCQ + Numerical) | Note |
|---|---|---|
| JEE Main 2024 | 43 | Highest of the three years |
| JEE Main 2025 | 23 | Notable dip |
| JEE Main 2026 | 31 | 6.53% weightage, up 34.92% year-on-year |
ExamSIDE's archive shows 331 total JEE Main questions from this chapter since 2002, across 180 papers — split 250 single-correct MCQ and 80 Numerical, with a rare single multi-correct MCQ appearing in the archive's older AIEEE-era papers. The 2024 spike (43 questions) followed by the 2025 dip and 2026 partial recovery suggests some year-to-year volatility in exactly how many sessions draw from this chapter — but the weightage trend into 2026 (up 34.92%) signals NTA is leaning into it more, not less, going forward.
| Year | Questions | Note |
|---|---|---|
| 2020 | 3 | MCQ Multi + Numerical |
| 2021 | 4 | Highest of the window |
| 2022 | 2 | MCQ Multi + Numerical |
| 2023 | 2 | MCQ Single + Numerical |
| 2024 | 3 | MCQ Single + Numerical (×2) |
| 2025 | 1 | Numerical only |
| 2026 | 2 | MCQ Single + MCQ Multi |
ExamSIDE's full archive (1978–2026) records 113 total JEE Advanced questions across 62 papers, including a large historical share of Subjective-format questions (27) from the pre-2000s era before the exam moved to its current MCQ/Numerical format. Since 2020, the chapter has shown up in some form every single year — smaller than Matrices and Determinants' consistent 2+ per year, but a real, ongoing presence rather than a chapter you can deprioritise for Advanced.
| Sub-topic | JEE Main (2024–26) | JEE Advanced (2020–26) | Combined |
|---|---|---|---|
| Direct Evaluation (Substitution, By Parts & Properties) | 76 | 8 | 84 |
| Greatest Integer & Piecewise/Modulus Integrals | 14 | 2 | 16 |
| Functional & Differentiated Integral Equations | 5 | 7 | 12 |
| Definite Integral as Limit of a Sum | 1 | 0 | 1 |
1 of the 97 JEE Main questions couldn't be confidently sub-topic-classified — a small gap compared to the other chapters in this series, largely because integral notation (∫, dx, limits) survives text extraction far better than matrix entries or chemical structures do. Where we found real signal, we used it.
Direct Evaluation — substitution, integration by parts, and the "properties of definite integrals" (king's rule, odd/even symmetry) — dominates at 84 of 113 categorised combined questions, 74%. This single number should reshape your study priorities: fluency in recognising which technique applies to a given integral, especially the properties-based shortcuts, is worth more practice time than any other single skill in this chapter. Functional and differentiated integral equations are the clear second priority, and notably make up a much larger share of Advanced questions (7 of 17, 41%) than Main ones (5 of 97, 5%) — a real format difference worth knowing about.
Our Maths HOD, AKS Yadav Sir, drills technique-recognition specifically — spotting which of substitution, by-parts, or a properties shortcut applies before you write a single line of working.
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"The single biggest time-waster I see is students trying substitution first on every integral, even when the bounds and symmetry are screaming 'use king's rule.' Definite Integration rewards pattern recognition more than raw calculation skill — if you can identify the right property in five seconds, a question that would take three minutes of substitution work becomes a thirty-second answer." — AKS Yadav Sir, HOD Mathematics, JEE Prep Master
Across 2020–2026, a striking 41% of JEE Advanced questions in this chapter define a function f(x) implicitly through an integral equation involving f itself — "f(x) = eˣ + ∫(y+xeˣ)f(y)dy" style setups. Main almost never tests this format at this frequency (just 5% of its questions), meaning Advanced-specific prep genuinely needs separate, targeted practice here.
8 of the 14 Main "Greatest Integer & Modulus" questions in our 2024–2026 window came from 2026 alone — more than 2024 and 2025 combined. Questions like "∫(x^(1/3)-[x^(1/3)])dx" require you to split the integration interval at every point where the greatest-integer function jumps, then integrate piece by piece.
The 2026 28th January Evening Shift question first defines g(x) as an integral of f(t), then asks for local minima/maxima points of g, then asks for |p+q|. You're not being tested on integration in isolation — you're tested on using an integral's result as an input to an entirely separate calculus step (finding critical points).
The raw question count actually fell from 43 (2024) to 23 (2025) before the weightage figure climbed for 2026 — a reminder that a single down year doesn't mean a chapter is losing importance. Question count and percentage weightage don't always move together, since weightage also depends on how many total questions the whole paper carries that session.
Only 1 of 97 Main questions in our window used the definite-integral-as-limit-of-a-sum technique directly — but it still appeared, meaning you can't skip this topic entirely even though it's clearly the lowest-priority sub-topic by volume.
The property every JEE student needs automatic is King's Rule: ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx. This single substitution — replacing x with (a+b-x) throughout the integrand while keeping the same limits — is what makes symmetric-looking integrals (especially trig ratios over [0,π/2] like sinⁿx/(sinⁿx+cosⁿx)) collapse to a value you can find without ever doing the "real" integration. Also know: if f is odd, ∫₋ₐᵃ f(x)dx = 0; if f is even, ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx. For non-symmetric integrals, the standard toolkit (u-substitution, integration by parts using the LIATE priority order for choosing u) still applies exactly as in indefinite integration — the only new step is evaluating at the bounds afterward.
Worked example: Evaluate ∫₀^(π/2) sinⁿx/(sinⁿx+cosⁿx) dx for any n. Apply King's Rule with a=0, b=π/2: replacing x with (π/2-x) turns sin into cos and cos into sin, giving ∫₀^(π/2) cosⁿx/(cosⁿx+sinⁿx) dx — call the original integral I and this new one I as well (same value, different-looking integrand). Adding the original and transformed integrals: I + I = ∫₀^(π/2) [sinⁿx+cosⁿx]/[sinⁿx+cosⁿx] dx = ∫₀^(π/2) 1 dx = π/2. So 2I=π/2, meaning I=π/4 — regardless of what n actually is. This exact "add the original to its King's Rule transform" move resolves a huge fraction of the properties-based questions in this sub-topic.
The core technique: identify every point within the integration interval where the greatest-integer function [f(x)] or the expression inside a modulus changes value, split the integral into separate pieces at those points, and integrate each piece with its own simplified (constant or sign-fixed) integrand. For |x²-3x+2| type integrals, first factor and find where the expression changes sign (the roots), then flip the sign of the integrand on whichever sub-interval makes it negative before integrating normally.
Worked example: Evaluate ∫₀² |x-1|-x| dx. First find where each modulus expression changes sign within [0,2]: x-1 changes sign at x=1. Inside [0,1], x-1<0 so |x-1|=1-x; inside [1,2], x-1≥0 so |x-1|=x-1. Split the integral at x=1: ∫₀¹(1-x)-x dx + ∫₁²(x-1)-x dx. Each piece is now a plain polynomial integral with no absolute value left — simplify inside each piece first (combining like terms), then integrate normally and add the two results. The entire difficulty in this sub-topic is almost always in correctly identifying breakpoints and the correct sign on each resulting sub-interval; once that's done, the actual integration is routine.
When a function is defined via its own integral — f(x) = (known term) + ∫(kernel involving f(t)) dt — the standard approach is to differentiate both sides with respect to x using the Leibniz rule, which turns the integral equation into an ordinary differential equation you can solve with standard techniques. When F(x) is defined as ∫₀ˣ t·f(t) dt and you're told F(x²) in terms of x, differentiate F(x²) with respect to x using the chain rule (this brings in F'(x²)·2x) and match it against the known form of F' from the original definition.
Worked example: If F(x)=∫₀ˣ t f(t) dt and F(x²)=x⁴+x⁵, find f(r²) summed for r=1 to 12. Differentiate both sides of F(x²)=x⁴+x⁵ with respect to x: by the chain rule, d/dx[F(x²)] = F'(x²)·2x. By the Leibniz rule, F'(u)=u·f(u), so F'(x²)=x²f(x²), giving the left side as 2x³f(x²). The right side's derivative is 4x³+5x⁴. Equating: 2x³f(x²)=4x³+5x⁴, so f(x²)=2+2.5x. Substitute x²=r² (so x=r) to get f(r²)=2+2.5r, then sum this expression for r=1 to 12 using the standard arithmetic series sum formula. This Leibniz-differentiate-then-match approach is the single reusable technique behind nearly every question in this sub-topic.
Once you can spot it in five seconds, entire categories of "hard" integrals become fast marks. AKS Yadav Sir drills exactly this recognition speed in every session.
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| Concept | Formula / Rule |
|---|---|
| King's Rule | ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx |
| Odd function over symmetric limits | ∫₋ₐᵃ f(x)dx = 0 (f odd) |
| Even function over symmetric limits | ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx (f even) |
| Leibniz rule (differentiating an integral) | d/dx ∫ₐ^(g(x)) f(t)dt = f(g(x))·g'(x) |
| Sum-to-limit (Riemann sum) | lim(n→∞) (1/n)Σf(r/n) = ∫₀¹ f(x)dx |
| Common trig integral (King's Rule result) | ∫₀^(π/2) sinⁿx/(sinⁿx+cosⁿx) dx = π/4 |
The clearest shift is format concentration: Advanced leans heavily on functional/differentiated integral equations (41% of its recent questions) in a way Main simply doesn't. The 2021 Paper 2 pair of Numerical questions (both built around the same f, g, h function-triple with derivative conditions) shows Advanced's tendency to build one elaborate function-definition scenario and extract two or three separate numerical answers from it — getting more mileage out of a single setup than Main's typically self-contained questions. If you're Advanced-focused, dedicate specific practice time to Leibniz-rule-based functional equations rather than assuming your Main-style substitution/properties fluency automatically transfers.
Against Matrices and Determinants (steady, procedural, low-drama), Definite Integration is more volatile — a real dip-then-recovery in Main question counts across 2024-2026, and a 34.92% weightage swing into 2026. Against Sequences and Series (where Advanced weightage collapsed to zero), Definite Integration shows the opposite Advanced story: down in weightage percentage but still present every year since 2020, with a distinct format specialty (functional equations) that rewards dedicated Advanced-specific practice. Of every chapter in this series so far, this is the one where technique-recognition speed — not formula memorisation — is most directly tied to your score, since 74% of the marks come down to correctly identifying substitution vs. by-parts vs. a properties shortcut in the first few seconds of reading the question.
Week 1 — Direct evaluation foundations: Days 1–3: u-substitution and integration by parts (LIATE) refreshed specifically for definite bounds. Days 4–7: King's Rule and odd/even symmetry properties, drilled on real 2024–2026 trig-ratio integrals until pattern recognition is instant.
Week 2 — Greatest integer & modulus: Days 1–3: Splitting integration intervals at sign-change or integer-jump points. Days 4–7: Full practice set on greatest-integer integrals specifically, given their sharp rise in 2026 Main papers.
Week 3 — Functional & differentiated integral equations: Days 1–4: Leibniz rule for differentiating integrals with variable bounds. Days 5–7: Practice converting integral equations into solvable differential equations — this week matters disproportionately if you're targeting Advanced.
Week 4 — Limit-of-sum and full drilling: Days 1–2: Definite integral as a limit of a Riemann sum (low priority by volume, but not skippable). Days 3–5: Full timed sets from JEE Main 2024–2026. Days 6–7: JEE Advanced 2020–2026, with extra weight on functional-equation questions.
Knowing every formula doesn't help if you pick the slow path under exam pressure. AKS Yadav Sir's sessions specifically train fast, correct technique selection.
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Every mistake above is one AKS Yadav Sir sees repeated every session, and corrects before it costs you marks on an actual paper.
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For anyone working out how to study Definite Integration for JEE, weightage is the first signal — yes, at 6.53% weightage in 2026 (up 34.92% year-on-year), it's a fast-growing, high-return Calculus chapter with 97 questions across just the last three years.
Based on our 2024–2026 analysis: 43 questions in 2024, 23 in 2025, and 31 in 2026 — a real dip-then-partial-recovery pattern worth knowing about rather than assuming steady growth.
Yes, and consistently so — it's appeared every year since 2020, with a distinct emphasis on functional/differentiated integral equations that's worth separate, dedicated practice.
Direct evaluation techniques — substitution, integration by parts, and especially King's Rule/symmetry properties — account for 74% of all combined PYQs in this chapter, more than any other sub-topic by a wide margin.
Matrices rewards precise, error-free execution of known procedures. Definite Integration rewards fast pattern recognition — correctly identifying which technique (substitution, by-parts, or a properties shortcut) applies within the first few seconds, since the actual calculation is often short once you've picked the right approach.
You need it — many of the trig-ratio integrals in this chapter (like sinⁿx/(sinⁿx+cosⁿx) over [0,π/2]) don't have a clean direct antiderivative for general n, and King's Rule is the only practical route to a fast, correct answer.
Given its rising Main weightage and consistent Advanced presence, treat it as a top-priority Calculus chapter — a full 4 weeks if starting fresh, or 5–6 focused days for pure revision with extra time on properties-based technique drilling.
Yes — 8 of the 14 Main "Greatest Integer & Modulus" questions in our 2024–2026 window came from 2026 alone, a sharp recent increase worth reflecting in your practice balance.
Definite Integration is a chapter where one skill — fast, correct technique selection — explains the overwhelming majority of the marks. This guide was built by pulling every JEE Main question from 2024–2026 and every JEE Advanced question from 2020–2026 directly from ExamSIDE's chapter-wise archives, classifying each by the technique it actually rewards, and cross-checking our counts against ExamSIDE's own published totals (331 JEE Main questions since 2002, 113 JEE Advanced questions since 1978). Only 1 of 97 JEE Main questions couldn't be confidently classified — the smallest gap of any chapter in this series so far, thanks to how well integral notation survives text extraction compared to matrices or chemical structures. If there's one takeaway on how to study Definite Integration for JEE efficiently, it's this: master King's Rule and symmetry properties before anything else, since they alone unlock the majority of this chapter's marks.
