Definite Integration
Maharashtra Board · Class 12 · Mathematics & Statistics
Quick revision notes for Definite Integration — Maharashtra Board Class 12 Mathematics & Statistics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Fundamental Theorem of Integral Calculus
- If f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x)dx = F(b) - F(a)
- The constant of integration cancels out in definite integrals
- Notation: ∫ₐᵇ f(x)dx = [F(x)]ₐᵇ where 'a' is lower limit and 'b' is upper limit
Standard Integration Techniques for Definite Integrals
- Apply integration by parts: ∫udv = [uv]ₐᵇ - ∫vdu evaluated from a to b
- Use substitution method with appropriate limit changes
- Apply partial fraction decomposition for rational functions
Properties of Definite Integrals
- Property 1: ∫ₐᵃ f(x)dx = 0 (integral from a point to itself is zero)
- Property 2: ∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx (reversing limits changes sign)
- Property 3: ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx (additivity property)
Advanced Problem-Solving Techniques
- Parametric evaluation: Find unknown parameters using given integral values
- Comparison method: Compare integrals to establish inequalities or equalities
- Splitting integrals at points of discontinuity or where integrand changes form
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