∫ CBSE · Class 12 · Mathematics · Chapter 7

Chapter 7
Integrals

Complete chapter resources for CBSE Class 12 Maths — integration techniques, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

5Topics
10–14Board marks
8Sample questions
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Key Formulas — Chapter 7
  • Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
  • Substitution: ∫ f(g(x))·g'(x) dx = F(g(x)) + C
  • By parts: ∫ u·v dx = u·∫v dx − ∫(u'·∫v dx) dx
  • Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a)
  • Even/Odd property: ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx [f even] or 0 [f odd]
  • Symmetry property: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx
Chapter Overview

What this chapter covers

Integration is the reverse process of differentiation and is one of the two pillars of calculus. Chapter 7 introduces the indefinite integral (antiderivative) of a function and builds the toolkit students need to evaluate a wide variety of integrals — from straightforward standard forms to composite expressions requiring substitution, partial fractions, or integration by parts.

The chapter progresses from direct standard results (such as ∫sin x dx = −cos x + C and the power rule) through integration by substitution, integration using partial fractions (for rational functions), and integration by parts using the ILATE priority rule. Each method expands the class of functions that can be integrated analytically, and students are expected to recognise which technique applies at a glance — a skill that board examiners consistently test.

Definite integrals are introduced via the Fundamental Theorem of Calculus and extended with a rich set of properties — particularly the substitution property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx and the even/odd symmetry rules — that allow elegant evaluation of integrals whose antiderivatives are hard or impossible to write in closed form. Board questions on definite integrals using these properties appear every year and are among the highest-scoring single questions in the paper.

Topics

What's inside Chapter 7

As per NCERT Class 12 Mathematics Part II (CBSE syllabus)

Topic 1
Introduction & Standard Integrals
Integral as antiderivative. Geometrical meaning of the indefinite integral. Standard results: power rule, trigonometric, exponential, and logarithmic forms. Comparison of differentiation vs integration.
Topic 2
Integration by Substitution
Substituting a suitable variable to simplify the integrand. Key forms: ∫f(ax+b)dx, ∫tan x, ∫cot x, ∫sec x, ∫cosec x, and integrals of the form ∫[f'(x)/f(x)]dx = ln|f(x)| + C.
Topic 3
Integration by Partial Fractions
Decomposing rational functions P(x)/Q(x) into partial fractions based on the nature of factors of Q(x): linear distinct, linear repeated, and irreducible quadratic factors. Covers all NCERT standard cases.
Topic 4
Integration by Parts
The ILATE rule for selecting the first function. Standard applications: ∫x·eˣ dx, ∫x·sin x dx, ∫ln x dx, and the special result ∫eˣ[f(x) + f'(x)]dx = eˣ f(x) + C.
Topic 5
Definite Integrals & Properties
Fundamental Theorem of Calculus. Evaluation of definite integrals. Eight standard properties — including the substitution property, even/odd symmetry, and the 2a-period property — with applications to board-style problems.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 5 · Continuity & Differentiability
Derivatives of all standard functions used in reverse for integration
Ch 6 · Application of Derivatives
Chain rule and product rule — both underpin substitution and integration by parts
Chapter 7 Integrals
Leads to
Ch 8 · Application of Integrals
Area under curves and between curves — direct application of definite integrals
Ch 9 · Differential Equations
Solving separable and linear ODEs requires integration techniques from this chapter
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 Maths board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Identify correct standard result, nature of integral, or apply a property
Very Short Answer 2 1 Evaluate a direct substitution or standard-form integral
Short Answer 3 1–2 Integration by parts, partial fractions, or definite integral with one property
Long Answer 5 1 Multi-step definite integral using properties, or a combination of techniques
Total (approximate) 10–14 4–6 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 7. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The value of ∫ (1/x) dx is: (a) x + C (b) ln x + C (c) ln|x| + C (d) 1/x² + C
Q2 Easy 2 marks Short Answer
Evaluate: ∫ (2x + 3)⁵ dx
Q3 Medium 3 marks Short Answer
Evaluate: ∫ x · eˣ dx using integration by parts.
Q4 Medium 3 marks Short Answer
Find: ∫ (3x + 1) / [(x − 1)(x + 2)] dx using partial fractions.
Q5 Medium 3 marks Short Answer
Evaluate the definite integral: ∫₀^(π/2) sin²x dx
Q6 Hard 5 marks Long Answer
Evaluate: ∫₀^π x · sin x / (1 + cos²x) dx Use the property ∫₀^a f(x) dx = ∫₀^a f(a − x) dx to simplify before integrating.
Q7 Hard 5 marks Long Answer
Find: ∫ √(1 − sin 2x) dx, and hence evaluate ∫₀^(π/4) √(1 − sin 2x) dx. (Hint: write 1 − sin 2x = (cos x − sin x)² and consider the sign of cos x − sin x on [0, π/4].)
Q8 Hard 4 marks Word Problem
A function f satisfies f'(x) = 3x² − 4x + 1 and f(0) = 5. (i) Find f(x). (ii) Find the value of f(2) and f(−1). (iii) Evaluate ∫₀² f(x) dx.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Integrals chapter.

Board 20235 marks
Evaluate: ∫₀^π x / (1 + sin x) dx (All India 2023)
Board 20223 marks
Find: ∫ (x² + x + 1) / (x² + 1)(x + 2) dx using partial fractions. (Delhi 2022)
Board 20204 marks
Evaluate: ∫₋π^π (cos²x) / (1 + aˣ) dx, where a > 0, a ≠ 1. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Integrals carry in the CBSE Class 12 board exam? +
Integrals (Chapter 7) typically carries 10–14 marks in the CBSE Class 12 Mathematics board examination. This includes questions on indefinite integrals, definite integrals, and integration techniques such as substitution and integration by parts. It is one of the highest-weightage chapters in the Calculus unit.
Which integration method should students prioritise for CBSE board exams? +
Students should be fluent in all standard methods: direct standard formulas, substitution, integration by parts (ILATE rule), and partial fractions. Board questions frequently test integration by parts (4–5 marks) and substitution-based problems. Definite integrals using properties (especially f(a+b−x) and even/odd function properties) are also high-probability board questions.
What are the most important properties of definite integrals for CBSE Class 12? +
The CBSE board frequently tests four properties: (1) integral from a to b equals negative of integral from b to a; (2) splitting the interval — integral from a to b = integral from a to c + integral from c to b; (3) the substitution property integral from 0 to a of f(x)dx = integral from 0 to a of f(a−x)dx; and (4) for even/odd functions, integral from −a to a simplifies to 2 times integral from 0 to a (even) or zero (odd). These appear in nearly every board paper.
Is the ILATE rule mandatory for integration by parts in CBSE exams? +
ILATE (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential) is a mnemonic to choose the first function in integration by parts: integral of u·v dx = u·integral(v dx) − integral(u'·integral(v dx))dx. It is not a theorem, but it gives the correct choice in most standard cases. CBSE examiners accept any correct choice of first function, but ILATE reliably avoids mistakes and is the standard approach taught in NCERT.
How do I generate a custom question paper for Integrals using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 7 (Integrals), set your preferred question-type mix (MCQ, short answer, long answer) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.