Even/Odd Function Integration
Definite Integration & Area
17
JEE Qs
8%
Hard
40
min
Always check for symmetric limits and the even/odd nature of the integrand first, as it can often simplify a complex definite integral to zero or a much simpler form instantly.
🧮 Key Formulas
✅ Key Points for JEE
- 1These properties are applicable ONLY when the limits of integration are symmetric, i.e., of the form [-a, a].
- 2Identifying even/odd nature of common functions (e.g., sin x is odd, cos x is even, x^n is even if n is even, odd if n is odd).
- 3The product/quotient of two even functions is even. The product/quotient of two odd functions is even. The product/quotient of an even and an odd function is odd.
- 4The sum/difference of two even functions is even. The sum/difference of two odd functions is odd. The sum/difference of an even and an odd function is generally neither even nor odd.
- 5If the integrand is a composite function, determine its overall even/odd nature by substituting -x. If it's a sum of functions, determine the nature of each term separately.
⚠️ Common Mistakes
- ✕Applying the even/odd property to non-symmetric limits of integration.
- ✕Incorrectly identifying the even/odd nature of a function, especially complex or composite functions (e.g., misclassifying sin(x^2) vs (sin x)^2).
- ✕Assuming the sum of an even and an odd function is always odd or always even; it's generally neither.
- ✕Forgetting that if f(x) = 0 for all x in [-a, a], it is both even and odd, but the property still yields 0.
📝 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 11 Maths Ch 2: Relations and Functions (Even and Odd Functions)
- Class 12 Maths Ch 7: Integrals (Properties of Definite Integrals)