Last Updated: May 2026
JEE Main Definite Integrals 2027 is one of the highest-yield Mathematics topics — historically 2–3 questions in every JEE Main paper. Mastery of the 10 properties of definite integrals lets you bypass full integration in 60–70% of problems. This guide covers all properties, key special integrals, advanced tricks and 35 practice problems.
Why Definite Integrals Are JEE-Critical
Indefinite integrals require full antiderivative computation. Definite integrals — once you know the properties — can be solved without computing the antiderivative at all. JEE testers love this distinction. Two-thirds of JEE Main definite-integral problems are property-based, not computation-based.
The 10 Properties of Definite Integrals
| # | Property | Use |
|---|---|---|
| P1 | ∫ab f(x) dx = ∫ab f(t) dt | Variable substitution |
| P2 | ∫ab f(x) dx = − ∫ba f(x) dx | Limit interchange |
| P3 | ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx | Limit splitting |
| P4 (King) | ∫ab f(x) dx = ∫ab f(a+b−x) dx | Most-used property |
| P5 | ∫0a f(x) dx = ∫0a f(a−x) dx | Special case of P4 |
| P6 | ∫02a f(x) dx = ∫0a f(x) dx + ∫0a f(2a−x) dx | Doubling interval |
| P7 | ∫02a f(x) dx = 2∫0a f(x) dx if f(2a−x)=f(x); 0 if f(2a−x)=−f(x) | Symmetry on [0, 2a] |
| P8 | ∫−aa f(x) dx = 2∫0a f(x) dx if f even; 0 if f odd | Odd/even symmetry |
| P9 | ∫0nT f(x) dx = n ∫0T f(x) dx if f periodic with period T | Periodicity |
| P10 | ∫aa+nT f(x) dx = n ∫0T f(x) dx if f periodic with period T | Periodicity (shifted) |
Worked Example 1 — King’s Property (P4)
Evaluate I = ∫0π/2 sin x / (sin x + cos x) dx.
Apply P5: I = ∫0π/2 sin(π/2 − x) / (sin(π/2 − x) + cos(π/2 − x)) dx = ∫0π/2 cos x / (cos x + sin x) dx.
Add: 2I = ∫0π/2 1 dx = π/2. So I = π/4. Done — without computing antiderivative.
Worked Example 2 — Odd/Even (P8)
Evaluate I = ∫−11 x³ cos x dx.
x³ is odd. cos x is even. Product = odd × even = odd. So I = 0 by P8. Five seconds, no computation.
Worked Example 3 — Periodicity (P9)
Evaluate I = ∫010π |sin x| dx.
|sin x| has period π. So I = 10 × ∫0π |sin x| dx = 10 × ∫0π sin x dx = 10 × [−cos x]0π = 10 × 2 = 20.
Special Integrals to Memorise
| Integral | Value |
|---|---|
| ∫0π/2 sinnx dx | Wallis: (n−1)(n−3)…/n(n−2)… × π/2 (if n even) |
| ∫0π/2 cosnx dx | Same as sinn |
| ∫0π/2 log(sin x) dx | −(π/2) log 2 |
| ∫0∞ e−ax² dx | (1/2)√(π/a) |
| ∫0π x sin x dx | π |
Advanced Trick: King’s Property + Repeated Substitution
For I = ∫0π x f(sin x) dx — use P5 to get I = π/2 × ∫0π f(sin x) dx. This is a JEE classic since 2015.
Definite Integrals as Limit of a Sum
∫ab f(x) dx = limn→∞ h Σr=1n f(a + rh), where h = (b−a)/n.
Used to convert summation series into integrals — a JEE Advanced staple.
Common Pitfalls
- Forgetting P8 sign convention for odd functions over symmetric interval
- Misapplying P4 when limits aren’t symmetric about midpoint (a+b)/2
- Not recognising periodicity (e.g., |sin x| has period π not 2π)
- Computing antiderivative when property would solve in 5 seconds
- Confusing improper integrals (where one limit is infinite) with bounded definite integrals
JEE 2-Minute Solution Strategy
- Look at limits first — do they hint at symmetry? 0-to-π/2, −a-to-a, 0-to-2a all flag properties
- Check the integrand for odd/even structure
- Try King’s property (P4/P5) before computing antiderivative
- If periodic, count periods and reduce
- Compute antiderivative only as last resort
35 Practice MCQs
[cg_quiz id=”jee-definite-integrals-2027″]
FAQ
Q1. Are definite integrals in JEE Main or just Advanced?
Both. JEE Main typically has 1–2 property-based questions. Advanced may have 1 multi-part problem combining definite integrals with summations or area-under-curve.
Q2. Which property is most-tested?
King’s Property (P4/P5) — appears in 60% of property-based JEE definite integral problems.
Q3. Should I memorise Wallis formula?
Yes — for ∫0π/2 sinnx dx, knowing the formula saves 90 seconds vs reduction-formula derivation.
Q4. Are improper integrals tested?
JEE Main — no. JEE Advanced — yes, but rarely (1 question every 3–4 years).
Q5. How do I know which property to apply?
Look at the limits: 0-to-π/2 → P5 (King’s). −a-to-a → P8 (odd/even). 0-to-large → P9 (period). Keep practising — pattern recognition develops fast.