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Maths · Class 12 · Integral Calculus

Definite Integration for JEE

Everything you need for this topic on one page: the ideas, the formulas, the traps, and free lectures that explain it well.

Last updated: 26 October 2026

Direct answer

Fundamental theorem, properties of definite integrals, King's property, periodic and even/odd functions, and limits as sums.

Fundamental theorem, properties of definite integrals, King's property, periodic and even/odd functions, and limits as sums.

Key ideas

Definite Integration: what you must understand

  • ∫ₐᵇ f(x) dx = F(b) − F(a)
  • King's property: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
  • Odd function over [−a, a] integrates to 0
  • Leibniz rule for derivative of an integral
Formula sheet

Must-know formulas for Definite Integration

  • ∫₋ₐᵃ f = 2∫₀ᵃ f if f even, 0 if odd
  • ∫₀^(nT) f = n∫₀ᵀ f for period T
  • Leibniz: d/dx ∫_{g(x)}^{h(x)} f(t) dt = f(h)h′ − f(g)g′
  • Limit of a sum: lim (1/n)Σ f(r/n) = ∫₀¹ f(x) dx
Print

One-page formula sheet

Printable Definite Integration formula sheet: formulas, key ideas and traps on one page. Save it as a PDF for offline revision.

Avoid

Common mistakes students make

  • Not splitting at points where |f| or [x] changes
  • Forgetting to change limits after substitution
  • Sign errors in Leibniz rule
For the exam

How this topic is asked in JEE

Appears in almost every JEE Main shift. One of the most frequent Maths topics. King's property solves many questions in one line.

Prerequisites: Indefinite Integration.

NCERT map

Where Definite Integration is in NCERT

Chapter names follow the current rationalised NCERT textbooks. Read these sections before solving PYQs.

ClassBookChapterSections JEE asks from
Class 12Mathematics Part IIIntegralsDefinite Integral; Fundamental Theorem of Calculus; Evaluation of Definite Integrals by Substitution; Some Properties of Definite Integrals
JEE Advanced level

A JEE Advanced-level example on Definite Integration

Question. Evaluate ∫₀^π x·sin²x/(1 + cos²x) dx

Show solution
  1. Let I = ∫₀^π x·sin²x/(1 + cos²x) dx. Apply King's property (x → π−x): since sin(π−x) = sinx and cos²(π−x) = cos²x, we get I = ∫₀^π (π−x)·sin²x/(1 + cos²x) dx
  2. Add both forms: 2I = π ∫₀^π sin²x/(1 + cos²x) dx. Write sin²x = 1 − cos²x and simplify the fraction to get 2/(1 + cos²x) − 1
  3. So 2I = π[2∫₀^π dx/(1 + cos²x) − π]. For the integral, multiply top and bottom by sec²x and substitute t = tanx
  4. The integral ∫₀^π dx/(1 + cos²x) splits at π/2. Each half gives (1/√2)·tan⁻¹(t/√2) evaluated from 0 to ∞, which equals π/(2√2). Total = π/√2
  5. Therefore 2I = π[2·(π/√2) − π] = π[π√2 − π] = π²(√2 − 1), giving I = π²(√2 − 1)/2

Answer: π²(√2 − 1)/2

This example was solved twice independently and both solutions agree.

Free lectures

Best free lectures for Definite Integration

Full-chapter free lectures picked from YouTube search results by views, from established JEE channels (mostly in Hinglish). Use the subtopic list to jump straight to the part you need (timestamps come from each video's own chapter list). Watch one, then solve PYQs the same day.

Practise

Where to practise for free

FAQs

Frequently asked questions

Is Definite Integration important for JEE Main 2027?

Appears in almost every JEE Main shift. It is part of the official 'Integral Calculus' unit in the JEE Main 2027 syllabus.

What are the most important formulas in Definite Integration?

∫₋ₐᵃ f = 2∫₀ᵃ f if f even, 0 if odd; ∫₀^(nT) f = n∫₀ᵀ f for period T; Leibniz: d/dx ∫_{g(x)}^{h(x)} f(t) dt = f(h)h′ − f(g)g′.

How should I study Definite Integration for JEE?

Understand the key ideas from NCERT or one full lecture, learn the formula sheet, then solve 25–30 past JEE questions on the topic. One of the most frequent Maths topics. King's property solves many questions in one line.

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