Chapter 7 — Integrals — is the biggest and most important chapter, carrying 10-12 marks. Covers indefinite integrals (methods: substitution, partial fractions, by parts) and definite integrals (fundamental theorem, properties).
Key Concepts
Standard Integrals
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1) | ∫(1/x) dx = ln|x| + C
∫eˣ dx = eˣ + C | ∫aˣ dx = aˣ/ln a + C
∫sin x dx = −cos x + C | ∫cos x dx = sin x + C
∫sec²x dx = tan x + C | ∫cosec²x dx = −cot x + C
∫sec x tan x dx = sec x + C | ∫cosec x cot x dx = −cosec x + C
∫1/√(1−x²) dx = sin⁻¹x + C | ∫1/(1+x²) dx = tan⁻¹x + C
∫eˣ dx = eˣ + C | ∫aˣ dx = aˣ/ln a + C
∫sin x dx = −cos x + C | ∫cos x dx = sin x + C
∫sec²x dx = tan x + C | ∫cosec²x dx = −cot x + C
∫sec x tan x dx = sec x + C | ∫cosec x cot x dx = −cosec x + C
∫1/√(1−x²) dx = sin⁻¹x + C | ∫1/(1+x²) dx = tan⁻¹x + C
Methods of Integration
1. Substitution
If integrand = f(g(x))·g'(x), substitute t = g(x)
Example: ∫2x·cos(x²) dx → let t = x² → ∫cos t dt = sin t + C = sin(x²) + C
Example: ∫2x·cos(x²) dx → let t = x² → ∫cos t dt = sin t + C = sin(x²) + C
2. Integration by Parts
∫u·v dx = u∫v dx − ∫[u’·∫v dx] dx
ILATE rule for choosing u: Inverse trig > Logarithmic > Algebraic > Trig > Exponential
(Choose u as the function that comes first in ILATE)
ILATE rule for choosing u: Inverse trig > Logarithmic > Algebraic > Trig > Exponential
(Choose u as the function that comes first in ILATE)
3. Partial Fractions
For P(x)/Q(x) where deg P < deg Q:
(x+1)/[(x−1)(x+2)] = A/(x−1) + B/(x+2)
Repeated: 1/(x−1)² = A/(x−1) + B/(x−1)²
Quadratic: (x+1)/(x²+1)(x−1) = A/(x−1) + (Bx+C)/(x²+1)
(x+1)/[(x−1)(x+2)] = A/(x−1) + B/(x+2)
Repeated: 1/(x−1)² = A/(x−1) + B/(x−1)²
Quadratic: (x+1)/(x²+1)(x−1) = A/(x−1) + (Bx+C)/(x²+1)
Special Integrals
∫1/(x²+a²) dx = (1/a)tan⁻¹(x/a) + C
∫1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)| + C
∫1/√(a²−x²) dx = sin⁻¹(x/a) + C
∫1/√(x²+a²) dx = ln|x + √(x²+a²)| + C
∫eˣ[f(x) + f'(x)] dx = eˣf(x) + C ← very useful!
∫1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)| + C
∫1/√(a²−x²) dx = sin⁻¹(x/a) + C
∫1/√(x²+a²) dx = ln|x + √(x²+a²)| + C
∫eˣ[f(x) + f'(x)] dx = eˣf(x) + C ← very useful!
Definite Integrals
Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b) − F(a) where F'(x) = f(x)
Key Properties:
∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx ← most useful property!
∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even; = 0 if f is odd
∫₀²ᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(2a−x) = f(x); = 0 if f(2a−x) = −f(x)
Key Properties:
∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx ← most useful property!
∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even; = 0 if f is odd
∫₀²ᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(2a−x) = f(x); = 0 if f(2a−x) = −f(x)
Quick Revision Points
- ILATE for by-parts: Inverse trig > Log > Algebraic > Trig > Exp
- ∫eˣ[f(x)+f'(x)]dx = eˣf(x) + C
- Partial fractions: degree of numerator must be less than denominator
- ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx — the most powerful property
- Even function: ∫₋ₐᵃ = 2∫₀ᵃ; Odd function: ∫₋ₐᵃ = 0
Chapter Navigation
Previous: Application of Derivatives Class 12 Notes
Next: Application of Integrals Class 12 Notes
Related Chapters in Class 12 Maths
- Application of Integrals Class 12 Notes
- Differential Equations Class 12 Notes
- Continuity and Differentiability Class 12 Notes
Practice What You Learned
Test yourself with our JEE Main Mock Test Set 1 to see how well you’ve mastered the concepts.