Properties of definite integrals: Class 12 practice
∫₋ₐᵃ f(x) dx = 0 when f is odd and 2∫₀ᵃ f(x) dx when f is even. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Definite integrals
The idea behind properties of definite integrals
∫₋ₐᵃ f(x) dx = 0 when f is odd and 2∫₀ᵃ f(x) dx when f is even. ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx is useful for symmetric integrals.
Step-by-step method
- Check if the integrand is odd (f(−x) = −f(x)) or even (f(−x) = f(x)).
- Apply the property.
- Integrate the simpler form.
Worked example 1 (Basic)
Evaluate ∫ from −2 to 2 of (x^5 + 1x) dx.
- Integrand is odd
- = 0
- Answer: 0
Worked example 2 (Standard)
Evaluate ∫ from −3 to 3 of 4x² dx.
- Even: 2∫₀^3 4x² dx
- = 2 × 4 × 3³/3 = 72
- Answer: 72
Worked example 3 (Challenge)
Evaluate ∫ from −3 to 3 of (x^5 + 2x²) dx.
- Odd part integrates to 0
- Even part: 36
- Answer: 36
Common mistakes and misconceptions
- Calling x³ + x² odd
- Forgetting the factor 2 for even functions
Practice questions
- (Basic) Evaluate ∫ from −3 to 3 of (x^3 + 1x) dx.
- (Basic) Evaluate ∫ from −1 to 1 of (x^3 + 5x) dx.
- (Standard) Evaluate ∫ from −2 to 2 of 3x² dx.
- (Standard) Evaluate ∫ from −2 to 2 of 2x² dx.
- (Challenge) Evaluate ∫ from −2 to 2 of (x^5 + 4x²) dx.
- (Challenge) Evaluate ∫ from −1 to 1 of (x^3 + 3x²) dx.
Answer key with full working
- Q1: Integrand is odd → = 0 → Answer: 0
- Q2: Integrand is odd → = 0 → Answer: 0
- Q3: Even: 2∫₀^2 3x² dx → = 2 × 3 × 2³/3 = 16 → Answer: 16
- Q4: Even: 2∫₀^2 2x² dx → = 2 × 2 × 2³/3 = 32/3 → Answer: 32/3
- Q5: Odd part integrates to 0 → Even part: 64/3 → Answer: 64/3
- Q6: Odd part integrates to 0 → Even part: 2 → Answer: 2
Syllabus note
This skill is listed under "Integrals" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is properties of definite integrals in?
"Integrals", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.