Last Updated: May 2026
Chapter 7: Integrals is one of the highest-weight chapters of CBSE Class 12 Mathematics 2027 — typically worth 10-12 marks in the board exam across Section A (1-mark MCQs), Section B (2-mark short answers) and Section D (5-mark long answers). Mastering this chapter is the difference between a 95+ and a 88 in Maths because integration also underpins Application of Integrals (Chapter 8). This guide gives you the entire NCERT chapter compressed into one revision-ready document with worked methods and 30 important questions.
Snapshot — CBSE Chapter 7 Integrals
| Parameter | Detail |
|---|---|
| NCERT Chapter | Class 12 Maths Chapter 7 |
| Average Board Marks | 10-12 |
| Sub-topics | Indefinite, Definite, Substitution, By Parts, Partial Fractions |
| Most-asked technique | Integration by parts (∫u·v dx = u∫v − ∫u’·∫v) |
1. Indefinite Integral — Reverse of Differentiation
If d/dx [F(x)] = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration. Geometrically, ∫f(x)dx is a family of curves shifted vertically.
2. Standard Integrals (Memorise All)
| 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 |
| cosec x · cot x | −cosec x + C |
| 1/√(1−x²) | sin⁻¹ x + C |
| 1/(1+x²) | tan⁻¹ x + C |
3. Integration by Substitution
Replace the variable to simplify. Standard substitutions:
- For √(a² − x²) → x = a sin θ.
- For √(a² + x²) → x = a tan θ.
- For √(x² − a²) → x = a sec θ.
- For (ax+b)^n → t = ax+b.
4. Integration by Parts
∫u dv = uv − ∫v du. Choice of u via ILATE rule:
- I — Inverse trigonometric
- L — Logarithmic
- A — Algebraic
- T — Trigonometric
- E — Exponential
Special form: ∫e^x [f(x) + f'(x)] dx = e^x · f(x) + C.
5. Partial Fractions
For ∫P(x)/Q(x) dx where degree(P) < degree(Q):
- If Q(x) = (x−a)(x−b): P/Q = A/(x−a) + B/(x−b).
- If repeated linear factor (x−a)²: P/Q = A/(x−a) + B/(x−a)².
- If irreducible quadratic (x²+px+q): P/Q = (Ax+B)/(x²+px+q).
6. Definite Integrals
∫_a^b f(x)dx = F(b) − F(a), where F is any antiderivative.
Properties of Definite Integrals:
- ∫_a^b f(x) dx = ∫_a^b f(t) dt (variable doesn’t matter).
- ∫_a^b f(x) dx = −∫_b^a f(x) dx.
- ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx.
- ∫_a^b f(x) dx = ∫_a^b f(a+b−x) dx.
- ∫_0^a f(x) dx = ∫_0^a f(a−x) dx.
- ∫_0^(2a) f(x) dx = 2∫_0^a f(x) dx if f(2a−x) = f(x); = 0 if f(2a−x) = −f(x).
- ∫_(−a)^a f(x) dx = 2∫_0^a f(x) dx if even; = 0 if odd.
7. Worked Examples
Example 1. Evaluate ∫(2x + 3)/(x² + 1) dx.
Split: ∫2x/(x²+1) dx + ∫3/(x²+1) dx = ln(x²+1) + 3 tan⁻¹ x + C.
Example 2. ∫sin³ x dx.
sin³ x = sin x · sin² x = sin x · (1 − cos² x). Let u = cos x ⇒ du = −sin x dx.
∫sin x (1 − cos² x) dx = ∫(1 − u²)(−du) = u³/3 − u + C = cos³ x/3 − cos x + C.
Example 3. ∫x e^x dx (by parts).
u = x, dv = e^x dx ⇒ du = dx, v = e^x.
xe^x − ∫e^x dx = xe^x − e^x + C = e^x(x−1) + C.
Example 4. Evaluate ∫_0^(π/2) sin² x dx.
= ½ ∫_0^(π/2) (1 − cos 2x) dx = ½ [π/2 − (sin π)/2] = π/4.
Example 5. ∫1/(x² − 4) dx.
Partial fractions: 1/((x−2)(x+2)) = ¼ [1/(x−2) − 1/(x+2)].
Integral = ¼ ln|(x−2)/(x+2)| + C.
8. 30 Important Board-Exam Questions — Sample of 10
Q1 (1 mark). ∫(1/x²) dx = ?
Ans: −1/x + C.
Q2 (1 mark). ∫sec² x dx = ?
Ans: tan x + C.
Q3 (2 marks). Evaluate ∫(x²+1)/x dx.
Ans: x²/2 + ln|x| + C.
Q4 (2 marks). Evaluate ∫sin x · cos x dx.
Ans: ½ sin² x + C OR −½ cos² x + C OR −¼ cos 2x + C.
Q5 (3 marks). Evaluate ∫x · ln x dx.
Ans: by parts u = ln x, dv = x dx; result = (x²/2) ln x − x²/4 + C.
Q6 (3 marks). Evaluate ∫1/(x² + 4x + 5) dx.
Complete square: x² + 4x + 5 = (x+2)² + 1. So integral = tan⁻¹(x+2) + C.
Q7 (5 marks). Evaluate ∫(2x+3)/√(x²−2x+2) dx.
Q8 (5 marks). Evaluate ∫_0^π x sin x dx.
By parts; result = π.
Q9 (4 marks). Evaluate ∫_0^(π/4) tan² x dx.
tan² x = sec² x − 1; integral = tan x − x at 0 and π/4 = 1 − π/4.
Q10 (5 marks). Use property ∫_0^a f(x) dx = ∫_0^a f(a−x) dx to evaluate ∫_0^(π/2) sin x / (sin x + cos x) dx.
Result = π/4.
9. Common Mistakes to Avoid
- Forgetting the constant of integration C in indefinite integrals (loses ½ mark).
- Mixing the limits when applying property 4 (a+b−x) or property 6 (even/odd).
- Not taking the modulus inside ln (e.g., ∫1/x dx must be ln|x|, not ln x).
- Errors in ILATE rule choice for integration by parts.
Frequently Asked Questions
How many marks does Integrals carry in CBSE Class 12 Maths?
Typically 10-12 marks across MCQs, short answers and a long-answer 5-mark question. Application of Integrals (Chapter 8) adds another 4-6 marks.
Is integration by parts always asked?
Yes. At least one 3 or 5-mark question on integration by parts appears every year. Common examples: x e^x, x ln x, x sin x.
Should I memorise standard integrals?
Yes — memorise all 12 standard formulas, all 7 properties of definite integrals, and the ILATE rule. Without them, every problem takes twice as long.
Best book for Class 12 Integrals?
NCERT Class 12 Mathematics Part II + RD Sharma Class 12 Volume 2 + ML Aggarwal for additional practice. Solve all NCERT exercises before moving to reference books.
Can I skip partial fractions?
No. Partial fractions appear in 1-2 questions every year, often as a 5-mark long answer. Solve every NCERT exercise problem under partial fractions.
