Integrals — Mathematics Class 12 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Integrals (Class 12) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Integrals (CBSE & HBSE)
Integrals is the largest chapter in Class 12, carrying 12-15 marks in CBSE. Topics include indefinite integrals (standard forms, substitution, by parts, partial fractions) and definite integrals with their properties.
Indefinite Integrals — Standard Forms and Substitution
Indefinite Integration
Definition: If d/dx[F(x)] = f(x), then ∫f(x)dx = F(x) + C F(x) is the antiderivative of f(x); C is the constant of integration.
Standard Integral Formulas
| ∫f(x)dx | Result | ||
|---|---|---|---|
| ∫xⁿ dx (n≠−1) | xⁿ⁺¹/(n+1) + C | ||
| ∫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 | ||
| ∫tan x dx | ln | sec x | + C |
| ∫cot x dx | ln | sin x | + C |
| ∫sec²x dx | tan x + C | ||
| ∫cosec²x dx | −cot x + C | ||
| ∫sec x tan x dx | sec x + C | ||
| ∫1/√(1−x²) dx | sin⁻¹x + C | ||
| ∫1/(1+x²) dx | tan⁻¹x + C | ||
| ∫1/√(x²+a²) dx | ln | x+√(x²+a²) | + C |
| ∫1/(x²−a²) dx | (1/2a)ln | (x−a)/(x+a) | + C |
| ∫√(a²−x²) dx | (x/2)√(a²−x²)+(a²/2)sin⁻¹(x/a)+C |
Integration by Substitution
Method: Put u = g(x), then du = g'(x)dx ∫f(g(x))g'(x)dx = ∫f(u)du = F(u) + C = F(g(x)) + C
Common substitutions:
- ∫sin x cos x dx: put u = sin x or use sin 2x/2
- ∫tan⁴x sec²x dx: put u = tan x
- ∫x/(1+x²) dx: put u = 1+x²
- ∫eˣ f(x)+f'(x)) dx = eˣ f(x) + C (special result)
Useful identities for integration:
- sin²x = (1−cos2x)/2
- cos²x = (1+cos2x)/2
- sin 2x = 2 sin x cos x
CBSE Tip: Always look for a function-derivative pair. If you can spot f(x) and f'(x) in the integrand, the result is often F(x) + C directly. The formula ∫[f(x)+f'(x)]eˣdx = eˣf(x)+C is very frequently tested.
Integration by Partial Fractions and by Parts
Integration by Partial Fractions
Used when integrating rational functions P(x)/Q(x).
Step 1: If degree(P) ≥ degree(Q), perform polynomial long division first. Step 2: Factor the denominator Q(x). Step 3: Decompose into partial fractions:
| Denominator factor | Partial fraction form |
|---|---|
| (x−a) | A/(x−a) |
| (x−a)² | A/(x−a) + B/(x−a)² |
| (x²+bx+c) irreducible | (Ax+B)/(x²+bx+c) |
Example: ∫(2x+3)/((x+1)(x+2)) dx
- (2x+3)/((x+1)(x+2)) = A/(x+1) + B/(x+2)
- 2x+3 = A(x+2) + B(x+1)
- x=−1: 1=A, A=1; x=−2: −1=−B, B=1
- ∫[1/(x+1) + 1/(x+2)]dx = ln|x+1| + ln|x+2| + C
Integration by Parts (IBP)
Formula: ∫u·v dx = u·∫v dx − ∫(u'·∫v dx) dx
ILATE Rule (choose u in this priority order): I — Inverse trigonometric L — Logarithmic A — Algebraic T — Trigonometric E — Exponential
Repeated IBP: Sometimes needed when the integral recurs: ∫eˣ sin x dx: Apply IBP twice to get I = eˣ sin x/2 − eˣ cos x/2 + C
Special integrals:
- ∫√(a²−x²) dx = (x/2)√(a²−x²) + (a²/2)sin⁻¹(x/a) + C
- ∫√(x²+a²) dx = (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C
- ∫√(x²−a²) dx = (x/2)√(x²−a²) − (a²/2)ln|x+√(x²−a²)| + C
CBSE Tip: For IBP, the ILATE rule determines which function is u. After selecting u, compute u' (differentiate) and ∫v dx (integrate). The formula comes from the product rule in reverse.
Definite Integrals and Their Properties
Definite Integral
Definition: ∫[a to b] f(x)dx = [F(x)]ₐᵇ = F(b) − F(a) (where F is the antiderivative of f)
Fundamental Theorem of Calculus
- Part 1: If F(x) = ∫[a to x] f(t)dt, then F'(x) = f(x)
- Part 2: ∫[a to b] f(x)dx = F(b) − F(a) (F is any antiderivative of f)
Properties of Definite Integrals
| Property | Formula |
|---|---|
| P1 | ∫[a to b]f(x)dx = −∫[b to a]f(x)dx |
| P2 | ∫[a to a]f(x)dx = 0 |
| P3 | ∫[a to b]f(x)dx = ∫[a to c]f(x)dx + ∫[c to b]f(x)dx |
| P4 | ∫[a to b]f(x)dx = ∫[a to b]f(a+b−x)dx |
| P5 | ∫[0 to a]f(x)dx = ∫[0 to a]f(a−x)dx |
| P6 | ∫[0 to 2a]f(x)dx = 2∫[0 to a]f(x)dx if f(2a−x)=f(x) |
| P7 | ∫[0 to 2a]f(x)dx = 0 if f(2a−x)=−f(x) |
| P8 | ∫[−a to a]f(x)dx = 2∫[0 to a]f(x)dx if f is even |
| P9 | ∫[−a to a]f(x)dx = 0 if f is odd |
Even function: f(−x) = f(x); examples: cos x, x², |x| Odd function: f(−x) = −f(x); examples: sin x, x³, tan x
Application of P5 (King's Rule): ∫[0 to π] x sin x/(1+cos²x) dx: Put I = ∫[0 to π] (π−x)sin x/(1+cos²x) dx Adding: 2I = π∫[0 to π] sin x/(1+cos²x) dx → solve
CBSE Tip: King's Rule (P4 or P5) and the even/odd property (P8/P9) are the most powerful tools for definite integrals. Always check if the integrand simplifies using P5 (replace x with a−x) before attempting substitution.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Integrals notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Integrals chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Integrals.