Last Updated: May 2026
CBSE Class 12 Mathematics Chapter 7 — Integrals is the heaviest chapter in NCERT Class 12 Maths, contributing 10-12 marks in the CBSE board exam every year. The 2026-27 CBSE syllabus continues with the post-rationalised Chapter 7 covering indefinite integrals, integration techniques (substitution, partial fractions, by parts), definite integrals, and properties of definite integrals. This guide gives chapter-wise NCERT solutions, important formulae, exam-focused tips, and 30 board-style practice MCQs and short-answer questions.
Quick Facts: CBSE Class 12 Integrals 2026-27
| Aspect | Detail |
|---|---|
| NCERT chapter | Class 12 Mathematics, Chapter 7 |
| Total exercises | 11 (7.1 to 7.11) + Misc |
| Marks weightage | 10-12 (Calculus block: 35 marks) |
| Difficulty | Moderate to Hard |
| Question types | 1-mark MCQ, 3-mark short, 5-mark long, case study (4 marks) |
| Most-tested zones | Integration by parts, definite integral properties, partial fractions |
Section 7.1-7.2: Indefinite Integral and Standard Formulae
| Function | Integral |
|---|---|
| xn, n ≠ −1 | xn+1/(n+1) + C |
| 1/x | ln |x| + C |
| ex | ex + C |
| ax | ax/ln a + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec²x | tan x + C |
| cosec²x | −cot x + C |
| sec x · tan x | sec x + C |
| 1/√(1−x²) | sin⁻¹ x + C |
| 1/(1+x²) | tan⁻¹ x + C |
Section 7.3: Integration by Substitution
If u = f(x), du = f'(x) dx, then ∫g(f(x)) f'(x) dx = ∫g(u) du.
Standard substitutions:
- √(a² − x²) → x = a sin θ
- √(a² + x²) → x = a tan θ
- √(x² − a²) → x = a sec θ
- (a − x)/(a + x) → x = a cos 2θ
Section 7.4: Integration by Partial Fractions
- Factor the denominator into linear/quadratic terms.
- Express as A/(x−a) + B/(x−b) + …; solve for A, B, …
- Then integrate each fraction separately.
Special case: Repeated linear factor (x−a)² gives A/(x−a) + B/(x−a)².
Section 7.5: Integration by Parts (LIATE Rule)
∫u dv = uv − ∫v du
LIATE order (priority for choosing ‘u’):
- L — Logarithmic (ln x)
- I — Inverse trig (sin⁻¹, tan⁻¹)
- A — Algebraic (x², x³)
- T — Trigonometric (sin x, cos x)
- E — Exponential (ex)
Special: ∫ex[f(x) + f'(x)] dx = ex·f(x) + C — recurring board exam favourite.
Section 7.6-7.8: Special Integrals
| Integral | Result |
|---|---|
| ∫dx/(x²−a²) | (1/2a) ln |(x−a)/(x+a)| + C |
| ∫dx/(a²−x²) | (1/2a) ln |(a+x)/(a−x)| + C |
| ∫dx/(a²+x²) | (1/a) tan⁻¹(x/a) + C |
| ∫dx/√(a²−x²) | sin⁻¹(x/a) + C |
| ∫dx/√(a²+x²) | ln |x + √(a²+x²)| + C |
| ∫dx/√(x²−a²) | ln |x + √(x²−a²)| + C |
| ∫√(a²−x²) dx | (x/2)√(a²−x²) + (a²/2)sin⁻¹(x/a) + C |
Section 7.9: Definite Integral as Limit of a Sum
∫ab f(x) dx = limn→∞ Σk=1n f(a + kh)·h, where h = (b−a)/n.
Important for proving fundamental theorem of calculus.
Section 7.10-7.11: Properties of Definite Integrals (HIGH-YIELD)
- ∫ab f(x) dx = ∫ab f(t) dt — variable change.
- ∫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.
- ∫−aa f(x) dx = 2∫0a f(x) dx if f even, 0 if f odd.
- ∫02a f(x) dx = 2∫0a f(x) dx if f(2a−x) = f(x), 0 if f(2a−x) = −f(x).
Top 5 Board Exam Question Types
- 5-mark long question: Evaluate ∫(x²+1)/[(x²+2)(x²+3)] dx using partial fractions.
- 5-mark question on properties: Show ∫0π/2 sin x/(sin x + cos x) dx = π/4.
- 3-mark integration by parts: ∫x·sin⁻¹x dx.
- Case study (4-mark): Real-life integration application — area, work done, displacement.
- 1-mark MCQ: Standard formula recall (∫sec²x dx = ?).
FAQ — CBSE Class 12 Integrals 2026-27
Q1. What is the LIATE rule and when is it used?
Q2. How many marks does Integrals carry in CBSE 2026 board exam?
Q3. Which property of definite integrals is most asked?
Q4. Is integration by parts more important than substitution?
Q5. Are NCERT exemplar problems sufficient?
Practice MCQs
Quiz data missing.
Related Reading
- Class 12 Continuity & Differentiability
- Class 12 Physics — Electric Charges and Fields
- Class 12 Chemistry — Chemical Kinetics
- Class 12 Biology — Human Reproduction
- CBSE Class 12 Mock Test
Bottom line: Integrals is non-negotiable for the 95+ aspirant. Memorise the formula table, master integration-by-parts (LIATE), and lock the 7 properties of definite integrals. Practice 50 NCERT + 50 board questions to become exam-ready.
