MathsMediumClass 12

Properties of Definite Integrals (King's Rule etc.)

Definite Integration & Area

JEE Qs

15%

Hard

min

Master identifying the appropriate property for simplification based on the integrand and limits, as direct integration is often impractical or impossible.

🧮 Key Formulas

∫[a, b] f(x) dx = ∫[a, b] f(t) dt

∫[a, b] f(x) dx = -∫[b, a] f(x) dx

∫[a, b] f(x) dx = ∫[a, c] f(x) dx + ∫[c, b] f(x) dx

∫[a, b] f(x) dx = ∫[a, b] f(a+b-x) dx (King's Rule)

∫[0, a] f(x) dx = ∫[0, a] f(a-x) dx

∫[-a, a] f(x) dx = 2 * ∫[0, a] f(x) dx (if f is an even function, i.e., f(-x) = f(x))

∫[-a, a] f(x) dx = 0 (if f is an odd function, i.e., f(-x) = -f(x))

∫[0, 2a] f(x) dx = ∫[0, a] f(x) dx + ∫[0, a] f(2a-x) dx

∫[0, 2a] f(x) dx = 2 * ∫[0, a] f(x) dx (if f(2a-x) = f(x))

∫[0, 2a] f(x) dx = 0 (if f(2a-x) = -f(x))

∫[0, nT] f(x) dx = n * ∫[0, T] f(x) dx (if f(x) is periodic with period T)

✅ Key Points for JEE

1King's Rule (∫[a, b] f(x) dx = ∫[a, b] f(a+b-x) dx) is the most frequently applied property, often used to eliminate complex terms or simplify trigonometric expressions.
2For integrals with symmetric limits [-a, a], immediately check for even or odd functions; this can drastically simplify the integral to 0 or 2 times the integral over [0, a].
3The property ∫[0, 2a] f(x) dx is a common variant; learn to identify when f(2a-x) = f(x) or f(2a-x) = -f(x) to simplify.
4Periodic function properties (∫[0, nT] f(x) dx = n * ∫[0, T] f(x) dx) are crucial for integrals over multiple periods, especially involving functions like sin(x), cos(x), or step functions.
5Often, problems require applying a property, then adding the original and transformed integrals to solve for 'I' (e.g., I + I = simplified_integral).

⚠️ Common Mistakes

✕Incorrectly identifying whether a function is even or odd, or misapplying the conditions for properties involving 2a.
✕Failing to recognize the specific form of the integrand and limits that prompt the use of a property, leading to complex direct integration.
✕Algebraic errors when combining integrals after applying a property, especially with fractions or trigonometric identities.

📝 Practice Questions

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Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

2025·NumericalHard

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

2025·MCQMedium

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 _______.

2025·NumericalMedium

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

2025·MCQMedium

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

2025·MCQHard

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 ,

2025·MCQMedium

NCERT Chapters

Class 12 Maths Ch 7: Integrals