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6.Definite Integrals
Class 12 • Mathematics • NCERT
Definite Integrals
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This lesson introduces definite integrals as a limit of sums and explains their evaluation using fundamental properties, as prescribed by NCERT Class 12.
Lesson Objectives
- Understand the concept of definite integrals.
- Learn properties of definite integrals.
- Evaluate definite integrals using standard results.
- Prepare for NCERT and board exam questions.
1. Meaning of Definite Integral
A definite integral represents a fixed numerical value obtained by integrating a function between given limits.
∫ab f(x) dx
2. Fundamental Theorem of Calculus
If F(x) is an antiderivative of f(x), then:
∫ab f(x) dx = F(b) − F(a)
3. Properties of Definite Integrals
∫aa f(x) dx = 0
∫ab f(x) dx = − ∫ba f(x) dx
∫ab [f(x) + g(x)] dx = ∫ab f(x) dx + ∫ab g(x) dx
∫ab f(x) dx = − ∫ba f(x) dx
∫ab [f(x) + g(x)] dx = ∫ab f(x) dx + ∫ab g(x) dx
4. Symmetry Property
If f(x) is even or odd, definite integrals can be simplified.
If f(x) is even:
∫−aa f(x) dx = 2 ∫0a f(x) dx
∫−aa f(x) dx = 2 ∫0a f(x) dx
If f(x) is odd:
∫−aa f(x) dx = 0
5. NCERT Solved Example
Evaluate: ∫01 (2x + 1) dx
= [x² + x]01
= (1 + 1) − 0
= 2
6. Important NCERT Notes
• Definite integrals give numerical values
• Limits of integration are essential
• Constant of integration is not used
• Used in area under curves and applications
• Limits of integration are essential
• Constant of integration is not used
• Used in area under curves and applications
Practice Questions (NCERT)
- Define definite integral.
- State the fundamental theorem of calculus.
- Find ∫12 x dx.
- Evaluate ∫0π sin x dx.
- What is ∫aa f(x) dx?
- State the symmetry property for odd functions.
- Is constant of integration used?
- Find ∫−aa x² dx.
- What does a definite integral represent?
- Is ∫ab f(x) dx always positive?
✅ Show Answer Key
- Integral between fixed limits
- ∫ f(x) dx = F(b) − F(a)
- 3/2
- 2
- 0
- Integral is zero
- No
- 2a³/3
- Numerical value
- No
© Aviate Learning – Definite Integrals (NCERT Class 12)