Class 12 · Chapter 7 · Calculus unit (35 of 80 marks)

Integrals Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 67 board-style questions.

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67 questions
18 multiple choice, 3 assertion–reason, 18 very short answer, 18 short answer, 6 long answer, 4 case study
About 28 hours to master

Calculus unit: 35 of 80 theory marks (Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations).

NCERT textbook: Chapter 7 on ncert.nic.in (PDF)

Revision notes

Integrals — revision notes

1. Standard integrals

$\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C (n \neq - 1)$ , $\int \frac{d x}{x} = \log | x | + C$ , $\int e^{x} d x = e^{x} + C$ , $\int a^{x} d x = \frac{a^{x}}{\log a} + C$ .
$\int \sin x = - \cos x$ , $\int \cos x = \sin x$ , $\int \sec^{2} x = \tan x$ , $\int \tan x = \log | \sec x |$ , $\int \sec x = \log | \sec x + \tan x |$ , $\int \csc x = \log | \csc x - \cot x |$ .

2. Special forms

$\int \frac{d x}{x^{2} + a^{2}} = \frac{1}{a} \tan^{- 1} \frac{x}{a}$ ; $\int \frac{d x}{x^{2} - a^{2}} = \frac{1}{2 a} \log | \frac{x - a}{x + a} |$ ; $\int \frac{d x}{a^{2} - x^{2}} = \frac{1}{2 a} \log | \frac{a + x}{a - x} |$ .
$\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \sin^{- 1} \frac{x}{a}$ ; $\int \frac{d x}{\sqrt{x^{2} \pm a^{2}}} = \log | x + \sqrt{x^{2} \pm a^{2}} |$ .
$\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a}$ ; $\int \sqrt{x^{2} \pm a^{2}} d x = \frac{x}{2} \sqrt{x^{2} \pm a^{2}} \pm \frac{a^{2}}{2} \log | x + \sqrt{x^{2} \pm a^{2}} |$ .
Quadratic $a x^{2} + b x + c$ in the denominator (or under a root): complete the square. For $\frac{p x + q}{a x^{2} + b x + c}$ : write $p x + q = A \frac{d}{d x} (a x^{2} + b x + c) + B$ .

3. Techniques

Substitution: look for a function and its derivative; change the limits in definite integrals.
Partial fractions (proper fractions): $\frac{A}{x - a} + \frac{B}{x - b}$ ; repeated factor $\frac{A}{x - a} + \frac{B}{(x - a)^{2}}$ ; irreducible quadratic $\frac{B x + C}{x^{2} + b x + c}$ . Divide first if the fraction is improper.
By parts: $\int u v d x = u \int v d x - \int (u^{'} \int v d x) d x$ ; choose $u$ by ILATE. $\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C$ .

4. Definite integrals

If $F^{'} = f$ and $f$ is continuous on $[a, b]$ : $\int_{a}^{b} f = F (b) - F (a)$ . If $A (x) = \int_{a}^{x} f (t) d t$ , then $A^{'} (x) = f (x)$ .

$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$ ; $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ .
$\int_{- a}^{a} f = 2 \int_{0}^{a} f$ if $f$ is even, $0$ if $f$ is odd.
$\int_{0}^{2 a} f = 2 \int_{0}^{a} f$ if $f (2 a - x) = f (x)$ , $0$ if $f (2 a - x) = - f (x)$ .
Split at points where a modulus or greatest-integer expression changes form.

Worked example 1

$\int \frac{3 x + 1}{x^{2} + 2 x + 5} d x$ : $3 x + 1 = \frac{3}{2} (2 x + 2) - 2$ , giving $\frac{3}{2} \log (x^{2} + 2 x + 5) - \tan^{- 1} \frac{x + 1}{2} + C$ .

Worked example 2

$\int x^{2} e^{x} d x = x^{2} e^{x} - 2 \int x e^{x} d x = e^{x} (x^{2} - 2 x + 2) + C$ .

Worked example 3

$I = \int_{0}^{π / 2} \frac{\cos^{4} x}{\sin^{4} x + \cos^{4} x} d x$ : using $x \to \frac{π}{2} - x$ and adding, $2 I = \frac{π}{2}$ , so $I = \frac{π}{4}$ .

Common errors

Forgetting $+ C$ in indefinite integrals, or keeping old limits after a substitution.
$\int \frac{d x}{2 x + 3} = \frac{1}{2} \log | 2 x + 3 |$ , not $\log | 2 x + 3 |$ .
Applying partial fractions to an improper fraction without dividing first.
Using $F (b) - F (a)$ when $f$ is discontinuous in $[a, b]$ .
Sign slips with $\int \sin x d x = - \cos x$ .

Board tips

For "property" questions write the property used — it carries a mark.
Check an indefinite integral by differentiating your answer.
Definite integrals as the limit of a sum and integrals of the type $\int (p x + q) \sqrt{a x^{2} + b x + c} d x$ are no longer in the syllabus.

Topics in this chapter: Integration as the inverse of differentiation · Integrals of special forms · Properties of definite integrals · Integration by parts · Integration by substitution · Definite integrals and the fundamental theorem of calculus · Integrals using trigonometric identities · Integration by partial fractions.

Formulas for this chapter: Chapter 7 on the Class 12 formula sheet (free PDF and A3 poster).

Integrals Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 206 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceIntegration by substitution

Which of these is $\int \frac{\log x}{x} d x$ for $x > 0$ ?

Show worked answer

Put $t = \log x$ , $d t = \frac{d x}{x}$ : $\begin{aligned} \int t d t & = \frac{t^{2}}{2} + C \\ = \frac{(\log x)^{2}}{2} + C \end{aligned}$ A1
Correct option: (a) $\frac{(\log x)^{2}}{2} + C$

Q2·1 mark·Multiple choiceDefinite integrals and their properties

The value of $\int_{0}^{π / 4} \sec^{2} x d x$ is

Q3·1 mark·Assertion–reasonDefinite integrals and their properties

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $\int_{- 1}^{1} x^{3} d x = 0$ .

Reason (R): For every continuous function $f$ , $\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerIntegration by substitution

Evaluate $\int \frac{2 x}{x^{2} + 4} d x$ .

Show worked answer

Answer:

\log (x^{2} + 4) + C

Put $t = x^{2} + 4$ , $d t = 2 x d x$ : $\int \frac{d t}{t}$ M1
$= \log (x^{2} + 4) + C$ A1

Q5·2 marks·Short answerIntegration by parts

Evaluate $\int x \cos 3 x d x$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerPartial fractions

Evaluate $\int \frac{3 x + 1}{(x - 1) (x + 3)} d x$ .

Show worked answer

Answer:

\log | x - 1 | + 2 \log | x + 3 | + C

$\frac{3 x + 1}{(x - 1) (x + 3)} = \frac{A}{x - 1} + \frac{B}{x + 3}$ , so $3 x + 1 = A (x + 3) + B (x - 1)$ M1
$x = 1$ : $4 = 4 A \Rightarrow A = 1$ ; $x = - 3$ : $- 8 = - 4 B \Rightarrow B = 2$ A1
Integral $= \log | x - 1 | + 2 \log | x + 3 | + C$ A1

Q7·3 marks·Short answerIntegration by substitution

Evaluate $\int_{0}^{π / 2} \frac{\sin x}{1 + \cos^{2} x} d x$ .

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerDefinite integrals and their properties

Evaluate $\int_{1}^{3} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4 - x}} d x$ .

Show worked answer

Answer:

1

Use $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$ with $a + b = 4$ M1
$I = \int_{1}^{3} \frac{\sqrt{4 - x}}{\sqrt{4 - x} + \sqrt{x}} d x$ A1
Adding the two forms: $\begin{aligned} 2 I & = \int_{1}^{3} \frac{\sqrt{x} + \sqrt{4 - x}}{\sqrt{x} + \sqrt{4 - x}} d x \\ = \int_{1}^{3} 1 d x \end{aligned}$ M1
$2 I = 3 - 1 = 2$ A1
$I = 1$ A1

Q9·5 marks·Long answerPartial fractions

Evaluate $\int \frac{x^{2} + x + 1}{(x + 1)^{2} (x + 2)} d x$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyDefinite integrals and their properties

Cycle on a straight track. A cyclist moves along a straight track with velocity $v (t) = 3 t^{2} + 2 t$ m/s, $t$ seconds after starting.

Write $\int v (t) d t$ . [1 mark]
How far does the cyclist travel in the first $2$ seconds? [1 mark]
The cyclist starts $5$ m from a marker. Find the distance from the marker at $t = 3$ and the average velocity over the first $3$ seconds. [2 marks]

Show worked answer

$t^{3} + t^{2} + C$ A1
$[t^{3} + t^{2}]_{0}^{2} = 8 + 4 = 12$ m A1
$s (3) = 5 + [t^{3} + t^{2}]_{0}^{3} = 5 + 36 = 41$ m M1
Average velocity $= \frac{36}{3} = 12$ m/s A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can integrate the standard forms, use simple substitutions and evaluate basic definite integrals.

Read first: 1. Standard integrals; 3. Techniques; 4. Definite integrals 15 practice questions · checkpoint: 4 questions, 6 marks, pass 80%

Practise step 1

Step 2

Board standard

You can use the special-form integrals, partial fractions and integration by parts on board-standard questions.

Read first: 2. Special forms; 3. Techniques; Worked examples 1-2 15 practice questions · checkpoint: 4 questions, 10 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write complete long answers on definite integrals, including the property questions, with each step and the property named.

Read first: 4. Definite integrals; Worked example 3; Board tips 7 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle the hardest property-based integrals and the tricky substitutions that decide 95+.

Read first: 4. Definite integrals; Common errors 6 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 5 of the 67 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceIntegration as the inverse of differentiation

$\int e^{2 \log_{e} x} d x$ (for $x > 0$ ) equals

(a) $2 x + C$
(b) $\frac{e^{2 \log x}}{2} + C$
(c) $x^{2} + C$
(d) $\frac{x^{3}}{3} + C$

Q2·1 mark·Multiple choiceIntegration by substitution

$\int \frac{d x}{x \log_{e} x}$ (for $x > 1$ ) equals

(a) $\log | \log x | + C$
(b) $\frac{(\log x)^{2}}{2} + C$
(c) $\frac{1}{\log x} + C$
(d) $x \log x + C$

Q3·2 marks·Very short answerIntegration by substitution

Find $\int \frac{\sin 2 x}{1 + \cos^{2} x} d x$ .

Worked solution·2 marks·Very short answerIntegration by parts

Find $\int x \sec^{2} 2 x d x$ .

Full worked solution Free, no sign-in

$= x \cdot \frac{\tan 2 x}{2} - \int \frac{\tan 2 x}{2} d x$ M1
$= \frac{x \tan 2 x}{2} - \frac{1}{4} \log | \sec 2 x | + C$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S130. All 24 →

Knowledge organiser: Integrals on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Integration by parts. All formulas A–Z

Where marks are lost in Integrals

Missing +C in indefinite integrals. Fix: write + C on every indefinite answer; the A1 is lost without it.
Keeping old limits after a substitution in a definite integral. Fix: change the limits in the same line as the substitution, or return to x before substituting limits.
Missing the 1/a factor: ∫dx/(2x + 3) = ½ log|2x + 3|. Fix: differentiate your answer mentally to check.
Partial fractions on an improper fraction. Fix: divide first when the degree of the numerator is not less than the denominator's.
Property questions without naming the property. Fix: write '∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx' before using it; it carries a mark.
Sign slips with ∫sin x dx = −cos x and in integration by parts. Fix: choose u by ILATE and write the formula ∫u v dx = u∫v − ∫(u′∫v) before substituting.
Studying definite integrals as a limit of a sum or ∫(px + q)√(ax² + bx + c)dx. Fix: both are out of the current syllabus.

Exam strategy: how answers are step-marked · competency-based questions

Integrals in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 12, 23, 27, 29, 34
Sample paper 2 (Board standard): questions 13, 14, 24, 27
Sample paper 3 (Above board): questions 11, 23, 28, 34
Sample paper 4 (Above board): questions 13, 14, 24, 27, 37
Sample paper 5 (95+ challenge): questions 11, 23, 28, 29

Want it against the clock? Take a timed 30-mark chapter test on Integrals, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on integrals, areas and differential equations.

Integrals: common questions

What is in CBSE Class 12 Maths Chapter 7, Integrals?

The topics are: Integration as the inverse of differentiation, Integrals of special forms, Properties of definite integrals, Integration by parts, Integration by substitution, Definite integrals and the fundamental theorem of calculus, Integrals using trigonometric identities, Integration by partial fractions. This page has revision notes for each, and 67 board-style practice questions with step-marked solutions.

How many marks is Integrals worth?

It is in the Calculus unit, which carries 35 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Integrals?

Missing +C in indefinite integrals. Fix: write + C on every indefinite answer; the A1 is lost without it. Keeping old limits after a substitution in a definite integral. Fix: change the limits in the same line as the substitution, or return to x before substituting limits.

Are the Integrals important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Integrals — revision notes

1. Standard integrals

2. Special forms

3. Techniques

4. Definite integrals

Worked example 1

Worked example 2

Worked example 3

Common errors

Board tips

Integrals Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Integrals

Integrals in our sample papers

Integrals: common questions

Work through the Route to 95 with your progress saved