CBSE PREP | APRIL 2026
Complete CBSE Class 12 Mathematics Chapter 7 — Integrals: all methods, standard formulas, step-by-step solved examples, and 10 practice MCQs for board exams 2027.
Chapter 7 (Integrals) typically contributes 5-8 marks to CBSE Class 12 Mathematics board exams. It also forms the foundation for Chapter 8 (Applications of Integrals) and is a key topic in JEE and other competitive exams. Mastering this chapter is non-negotiable for a 90+ score in Mathematics.
Last Updated: April 2026
Introduction to Integration — Anti-Differentiation
Integration is the reverse process of differentiation. If d/dx [F(x)] = f(x), then we write:
∫ f(x) dx = F(x) + C
where F(x) is the antiderivative (primitive) of f(x) and C is the constant of integration.
• F(x) is NOT unique — F(x) + C represents a family of antiderivatives
• The process of finding integrals is called integration or anti-differentiation
Standard Forms of Integrals — Formula Sheet
These standard integrals must be memorised for CBSE board exams:
| Function f(x) | Integral ∫f(x)dx |
|---|---|
| x^n (n ≠ -1) | x^(n+1)/(n+1) + C |
| 1/x | ln|x| + C |
| e^x | e^x + C |
| a^x | a^x / ln(a) + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| sec²(x) | tan(x) + C |
| cosec²(x) | -cot(x) + C |
| sec(x)tan(x) | sec(x) + C |
| 1/√(1-x²) | sin⁻¹(x) + C |
| 1/(1+x²) | tan⁻¹(x) + C |
Methods of Integration
1. Integration by Substitution:
When the integrand is of the form f(g(x))·g'(x), substitute t = g(x), then dt = g'(x)dx.
Example: ∫ 2x·sin(x²) dx — Let t = x², then dt = 2x dx
= ∫ sin(t) dt = -cos(t) + C = -cos(x²) + C
2. Integration by Parts (IBP):
Formula: ∫ u·v dx = u·∫v dx – ∫ [du/dx · ∫v dx] dx
L = Logarithmic functions (ln x, log x)
I = Inverse trigonometric functions (sin⁻¹x, tan⁻¹x)
A = Algebraic functions (x², x, polynomials)
T = Trigonometric functions (sin x, cos x)
E = Exponential functions (e^x, a^x)
Choose u as the function that appears EARLIER in the LIATE order. The other function is v.
3. Integration by Partial Fractions:
Used when integrating rational functions P(x)/Q(x) where degree of P(x) less than degree of Q(x). Decompose into partial fractions first, then integrate each term separately.
Example: ∫ 1/[(x+1)(x+2)] dx
Decompose: 1/[(x+1)(x+2)] = A/(x+1) + B/(x+2)
Solving: A = 1, B = -1
∫ [1/(x+1) – 1/(x+2)] dx = ln|x+1| – ln|x+2| + C = ln|(x+1)/(x+2)| + C
Definite Integrals — Properties and Applications
The definite integral ∫(a to b) f(x) dx represents the net area bounded by y = f(x), the x-axis, and lines x = a and x = b.
Important Properties of Definite Integrals (must know for CBSE):
- Reversal: ∫(a to b) f(x)dx = -∫(b to a) f(x)dx
- Zero: ∫(a to a) f(x)dx = 0
- Additivity: ∫(a to b) f(x)dx = ∫(a to c) f(x)dx + ∫(c to b) f(x)dx
- Even/Odd: If f(-x) = f(x) [even], ∫(-a to a) f(x)dx = 2∫(0 to a) f(x)dx; If f(-x) = -f(x) [odd], ∫(-a to a) f(x)dx = 0
S = Substitution — when you see f(g(x))·g'(x)
P = Parts (LIATE) — when you see a product of two different types of functions
Pa = Partial Fractions — when you see P(x)/Q(x) rational functions
“Solve Problems Patiently” — use S then P then Pa in that preference order
Common Mistakes to Avoid in Integration
- Forgetting the constant C in indefinite integrals (a common marks deduction in CBSE)
- Wrong sign: Remember ∫ sin(x) dx = -cos(x) + C (NEGATIVE cosine)
- Not checking if substitution limits need to be changed in definite integrals
- Applying integration by parts in wrong order (violating LIATE rule)
- Not factoring the denominator completely in partial fractions
Practice MCQs — CBSE Class 12 Mathematics Integrals
Practice Quiz — 10 CLAT-Style Questions
Click an option to reveal the answer and explanation.
Frequently Asked Questions — Class 12 Integrals
Q: What is the CBSE Class 12 exam pattern for Integrals chapter?
A: In the CBSE board exam, Integrals (Chapter 7) typically contributes to 5-8 marks through a mix of: 1-mark MCQ/Assertion-Reason (1-2 questions), 2-mark Very Short Answer, and 5-mark Long Answer questions. Chapter 8 (Applications of Integrals) adds another 5-6 marks.
Q: Is Integrals difficult in Class 12?
A: Integrals requires practice more than conceptual understanding. With daily practice of 5-10 integrals for 2 months, most students can master this chapter. The key is memorising standard formulas and recognising which method to apply.
Q: How to remember all integration formulas for board exams?
A: Make a formula sheet with all standard integrals and revise it daily. Create personal mnemonics. Practice derivation — if you can derive a formula, you won’t forget it.
Q: Which type of integration questions are asked in CBSE board exams?
A: CBSE typically tests: substitution method (most common), integration by parts (high marks questions), partial fractions (5-mark questions), and properties of definite integrals (a definite integral using properties is a classic board question).
Sources: NCERT Mathematics Part II (Class 12), CBSE Sample Paper 2025-26, Ready For Boards study material. Content accurate as of April 2026.
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