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Properties of Definite Integrals — All 9 properties

Definite Integration & Area

17

JEE Qs

8%

Hard

75

min

Master the King Property (P3/P4) and symmetry properties (P6/P7); they are the most common tools for simplifying definite integrals in JEE problems.

🧮 Key Formulas

P0: ∫[a,b] f(x) dx = ∫[a,b] f(t) dt
P1: ∫[a,b] f(x) dx = -∫[b,a] f(x) dx
P2: ∫[a,b] f(x) dx = ∫[a,c] f(x) dx + ∫[c,b] f(x) dx
P3 (King Property): ∫[a,b] f(x) dx = ∫[a,b] f(a+b-x) dx
P4 (Queen Property): ∫[0,a] f(x) dx = ∫[0,a] f(a-x) dx
P5: ∫[0,2a] f(x) dx = ∫[0,a] f(x) dx + ∫[0,a] f(2a-x) dx
P6: ∫[0,2a] f(x) dx = 2∫[0,a] f(x) dx if f(2a-x) = f(x)
P6: ∫[0,2a] f(x) dx = 0 if f(2a-x) = -f(x)
P7: ∫[-a,a] f(x) dx = 2∫[0,a] f(x) dx if f(x) is even (f(-x) = f(x))
P7: ∫[-a,a] f(x) dx = 0 if f(x) is odd (f(-x) = -f(x))
P8 (Periodicity): ∫[0,nT] f(x) dx = n * ∫[0,T] f(x) dx if f(x) is periodic with period T
P8 (Periodicity): ∫[a, a+T] f(x) dx = ∫[0, T] f(x) dx if f(x) is periodic with period T

✅ Key Points for JEE

  • 1The King Property (P3/P4) is arguably the most powerful and frequently used property, often simplifying complex integrands involving trigonometric functions or logarithms.
  • 2Always check for symmetry (even/odd functions) when limits are of the form [-a, a] or [0, 2a] as this can immediately reduce the integral to zero or simplify calculations significantly.
  • 3When facing piecewise functions or integrals with absolute values, use property P2 (splitting limits) at critical points to break down the integral into manageable parts.
  • 4For periodic functions, the value of the definite integral over any interval whose length is an integral multiple of the period can be simplified using P8.
  • 5Sometimes, combining two properties (e.g., P3 and then P7/P6) is required to solve an integral, or applying a property (like P3) and adding the original and new integral is key to evaluation.

⚠️ Common Mistakes

  • ✕Incorrectly identifying even/odd functions or f(2a-x) = f(x)/-f(x), leading to wrong application of P6/P7.
  • ✕Failing to recognize when to apply the King Property (P3/P4), especially when the integrand looks complex but simplifies drastically upon application.
  • ✕Errors in splitting limits (P2) for absolute value or greatest integer functions, often missing all critical points or sign changes.
  • ✕Not checking for periodicity when appropriate, particularly for trigonometric functions, which can simplify limits of integration significantly.

📝 Practice Questions

See all

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 Mathematics Ch 7: Integrals