NDA Maths · Teaching notes
Definite Integration — NDA Mathematics
Definite Integration is a high-yield, rising chapter in NDA Maths — 66 PYQs across 2017–2026, ~20% HARD, and built on a small set of powerful tricks rather than brute-force antidifferentiation. The chapter teaches in five movements: (1) Fundamental theorem, periodic integrals, and Leibniz rule — what a definite integral IS and the shortcuts for derivatives, periods, and variable limits; (2) Properties — symmetry, King's property, and odd/even — the heart of the chapter and its HARD pocket, where the 'add the integral to its own reflection' move evaluates integrals you could never antidifferentiate; (3) Integration of absolute value, piecewise, and greatest-integer functions — split at the break-points and integrate each piece; (4) Area under curves — the geometric reading of the integral; (5) Definite integrals in function conditions — recovering unknown coefficients from integral equations. 11 concepts, every PYQ tagged. This chapter assumes you can already find antiderivatives — for substitution, by-parts, and partial fractions, see the Indefinite Integration notes; here the focus is the definite-integral-specific machinery.
Subtopic notes
Fundamental Theorem, Periodicity and the Leibniz Rule
11 PYQsA definite integral is the change in an antiderivative across the limits; periodic integrands repeat over each period, and the Leibniz rule differentiates an integral with a variable limit.
Open note
Properties — King's, Symmetry and Standard Results
32 PYQsThe properties of definite integrals — above all King's property and odd/even symmetry — evaluate integrals you could never antidifferentiate, by adding an integral to its own mirror image.
Open note
Absolute Value, Piecewise and Greatest-Integer Integrals
17 PYQsWhen the integrand changes formula across the interval — an absolute value, a piecewise rule, or a greatest-integer function — split the integral at every break-point and integrate each piece on its own.
Open note
Area Under and Between Curves
3 PYQsArea is the integral of the gap between curves; use the absolute value (or split at intersections) so every piece counts positively, and exploit symmetry to halve the work.
Open note
Recovering a Function from Integral Conditions
3 PYQsWhen an unknown function has parameters and you are given several integral or derivative conditions, each condition becomes one linear equation — solve the system for the parameters.
Open note
PYQ weightage by concept
11 concepts · 66 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
11 concepts · 66 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| The Fundamental Theorem of Calculus | 6 | 9% |
| Integrals of periodic functions | 3 | 5% |
| The Leibniz rule — differentiating an integral | 2 | 3% |
| Concept | PYQs | Share |
|---|---|---|
| King's property — the reflection trick | 14 | 21% |
| Odd/even symmetry over a symmetric interval | 7 | 11% |
| Standard results and trig reductions | 6 | 9% |
| Direct evaluation — simplify, then integrate | 5 | 8% |
| Concept | PYQs | Share |
|---|---|---|
| Integrating absolute-value and piecewise functions | 10 | 15% |
| Integrating greatest-integer (floor) functions | 7 | 11% |
| Concept | PYQs | Share |
|---|---|---|
| Solving for coefficients from integral conditions | 3 | 5% |
Formula & revision sheet
3 formulas · 11 gotchas across all subtopics — the exam-eve cheat-sheet
Formula & revision sheet
3 formulas · 11 gotchas across all subtopics — the exam-eve cheat-sheet
Formulas (2)
Watch out for (3)
- Integrating a derivative is not always trivial→ The Fundamental Theorem of Calculus
- Use the function's period, not the trig argument's→ Integrals of periodic functions
- Don't forget the chain-rule factor→ The Leibniz rule — differentiating an integral
Watch out for (4)
- Add the reflected form — don't just substitute and stop→ King's property — the reflection trick
- Check parity of the WHOLE integrand→ Odd/even symmetry over a symmetric interval
- Reduce the power BEFORE integrating→ Standard results and trig reductions
- Transform the limits when you substitute→ Direct evaluation — simplify, then integrate
Watch out for (2)
- Find the zeros first — don't drop the absolute value→ Integrating absolute-value and piecewise functions
- ⌊x⌋ on [−1, 0) is −1, not 0→ Integrating greatest-integer (floor) functions
Watch out for (1)
- Area is unsigned — use the absolute value→ Area between curves
Watch out for (1)
- One condition, one equation — match the counts→ Solving for coefficients from integral conditions