By Abhishek Jaiswal, Mathematics faculty at Pramana Institute (IIT Roorkee, 18 years teaching JEE)
Definite integration is one of the highest-scoring chapters in JEE Main, and many of its questions are built so that no antiderivative is needed. If you try to integrate them directly, you lose ten minutes. One property, which students call the King property, turns them into two-line problems. Learn to spot when to use it, and these questions become the easiest marks in the paper.
The property
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx, and with a = 0: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
Why it is true: substitute t = a + b − x. Then dt = −dx, and the limits a and b swap places, which cancels the minus sign.
The trick in three steps
- Call the integral I.
- Replace x by (a + b − x) to get a second form of I.
- Add the two forms. If the integrand simplifies (often to a constant), you have 2I in one line.
Example 1: the classic ratio
I = ∫₀^(π/2) sinⁿx / (sinⁿx + cosⁿx) dx
Replacing x by π/2 − x swaps sin and cos, so I = ∫ cosⁿ x/(cosⁿ x + sinⁿ x) dx. Adding the two forms gives 2I = ∫₀^{π/2} 1 dx = π/2, so I = π/4 for every n.
Example 2: square roots
I = ∫₂³ √x / (√x + √(5 − x)) dx
Here a + b = 5. Replacing x by 5 − x gives √(5 − x)/(√(5 − x) + √x). Adding: 2I = ∫₂³ 1 dx = 1, so I = 1/2.
Example 3: x times a function of sin x
I = ∫₀^π x sin x / (1 + cos²x) dx
Replace x by π − x: sin x stays the same and cos²x stays the same, so I = ∫ (π − x) sin x/(1 + cos²x) dx. Adding: 2I = π ∫₀^π sin x/(1 + cos²x) dx. With t = cos x, this is π ∫₋₁¹ dt/(1 + t²) = π × π/2. So I = π²/4.
Bonus in one line
∫₀^{π/2} log(tan x) dx = 0, because the King property turns log(tan x) into log(cot x) = −log(tan x), so I = −I.
When to reach for the King property
- The integrand is a ratio like f(x)/(f(x) + f(a + b − x)).
- The limits are 0 to π/2 or 0 to π, with sin and cos (or tan and cot) in the integrand.
- There is a bare x multiplying a function that is unchanged when x becomes (a + b − x).
It won't help when the integrand changes into something unrelated after the swap. Then use substitution or parts as usual.
FAQs
Why is it called the King property? It is the most used property of definite integrals in JEE. "King" is a coaching nickname, not a textbook term.
Is this in the Class 12 NCERT book? Yes. It is one of the properties of definite integrals in the NCERT Class 12 Integrals chapter, so it counts for boards too.
Does it work for indefinite integrals? No. It depends on the limits, so it applies only to definite integrals.
Practise integration with weekly tests in a batch of at most 40 at Pramana, Geeta Bhawan Square, Indore. See our IIT JEE program or book free counselling.
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