King's Rule — ∫f(a+b-x) = ∫f(x)
Definite Integration & Area
17
JEE Qs
8%
Hard
60
min
Master the pattern recognition for problems where King's Rule simplifies integrands into standard forms, often leading to sums like 2I = constant or 2I = a simpler integral.
🧮 Key Formulas
✅ Key Points for JEE
- 1This property is primarily used to simplify definite integrals, especially when the integrand involves trigonometric functions (sin, cos, tan, cot, sec, cosec), inverse trigonometric functions, or logarithms.
- 2The standard strategy involves setting the original integral equal to 'I', applying King's Rule to obtain a new integral (which is also 'I'), and then adding the two integrals (leading to 2I) or, less commonly, subtracting them.
- 3Look for integrals with limits like [0, π/2], [0, π], [0, 2π], or any symmetric limits [a, b] where f(a+b-x) leads to a simpler or related expression.
- 4It is particularly effective for integrands of the form x * f(x) or expressions like log(1+tan x), log(sin x), log(cos x), or powers of sin/cos.
⚠️ Common Mistakes
- ✕Failing to recognize that applying King's Rule will simplify the integrand, or applying it when it doesn't lead to simplification.
- ✕Errors in algebraic manipulation, trigonometric identities, or logarithm properties after substituting (a+b-x) for x.
- ✕Incorrectly adding or subtracting the original and transformed integrals (e.g., forgetting to equate to 2I or miscalculating the resultant integral).
📝 Practice Questions
See allQ1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →
Q11.Let the area enclosed between the curves |y| = 1 −x2 and x2 + y2 = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β −γ| equals. (1) 27 (2) 33 (3) 15 (4) 18
Q21.If 24 ∫ 0 4 (sin 4x − 12π + [2 sin x])dx = 2π + α, where [⋅] denotes the greatest integer function, then α is equal to _______.
Q6. Let for f(x) = 7 tan8 x + 7 tan6 x −3 tan4 x −3 tan2 x, I1 = ∫π/40 f(x)dx and I2 = ∫π/40 xf(x)dx. Then 7I1 + 12I2 is equal to : (1) 2 (2) 1 (3) 2π (4) π
Q13.The area of the region, inside the circle (x −2√3)2 + y2 = 12 and outside the parabola y2 = 2√3x is : (1) 3π + 8 (2) 6π −16 (3) 3π −8 (4) 6π −8
Q7. The area of the region enclosed by the curves y = x2 −4x + 4 and y2 = 16 −8x is : (1) 8 (2) 4 3 3 (3) 8 (4) 5 x ∈R. Then the numbers of local maximum and local minimum points of f ,
NCERT Chapters
- Class 12 Maths Ch 7: Integrals