CBSE Class 12 Maths: Integrals — Complete Notes 2026

Basic Integrals

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫(1/x) dx = ln|x| + C
• ∫eˣ dx = eˣ + 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

Integration by Substitution

• Replace g(x) = t; then g'(x) dx = dt; transform integral into simpler form
• Example: ∫sin(x²) × 2x dx = ∫sin t dt = −cos t + C = −cos(x²) + C
• Standard: ∫f'(x)/f(x) dx = ln|f(x)| + C; ∫[f(x)]ⁿ f'(x) dx = [f(x)]ⁿ⁺¹/(n+1) + C

Integration by Parts

• ∫uv dx = u∫v dx − ∫(u')∫v dx dx; ILATE rule for choosing u: Inverse, Log, Algebraic, Trig, Exponential
• Example: ∫x eˣ dx: u = x, v = eˣ → x eˣ − ∫eˣ dx = x eˣ − eˣ + C = eˣ(x−1) + C
• Special cases: ∫eˣ[f(x) + f'(x)] dx = eˣ f(x) + C (memorise this formula)

Partial Fractions

• Decompose rational function (proper fraction) into simpler fractions before integrating
• Linear factors: A/(x−a) + B/(x−b); repeated: A/(x−a) + B/(x−a)²; irreducible quadratic: (Ax+B)/(x²+bx+c)
• Example: ∫1/[(x+1)(x+2)] dx = ∫[1/(x+1) − 1/(x+2)] dx = ln|x+1| − ln|x+2| + C

Definite Integrals

• ∫ᵃᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x) (Newton-Leibniz theorem)
• Properties: ∫ᵃᵇ f(x) dx = −∫ᵇᵃ f(x) dx; ∫ᵃᵃ f(x) dx = 0; ∫ᵃᵇ f(x) dx = ∫ᵃᶜ + ∫ᶜᵇ
• CBSE property: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx; even/odd function properties

Applications of Integrals

• Area between curve y = f(x) and x-axis from a to b: A = ∫ᵃᵇ |f(x)| dx
• Area between two curves: A = ∫ᵃᵇ [f(x) − g(x)] dx where f(x) ≥ g(x)
• CBSE: find intersection points first; determine which curve is above; set up integral correctly

CBSE Board Focus

• Integrals: 12–15 marks (largest single chapter in CBSE Class 12 Maths)
• Master all standard forms; practice definite integral properties questions
• Area questions: always draw diagram first; find intersection; use correct formula