Class 12 Maths Notes

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Integrals Class 12 Notes

Complete, exam-ready notes on integrals: the idea of integration as antiderivative, indefinite and definite integrals, substitution, integration by parts and partial fractions, and the fundamental theorem of calculus — written for CBSE boards and JEE revision.

Class:12Subject:MathematicsCovers:CBSE · JEE

6 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Integration reverses differentiation and measures exact area under a curve, using substitution, integration by parts and partial fractions plus the fundamental theorem of calculus.

Indefinite Integrals and Standard Forms

Definition

Antiderivative

An indefinite integral $\int f (x) d x = F (x) + C$ is the family of functions whose derivative is f(x); C is the constant of integration.

Power rule:
$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$
.
Standard forms:
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
,
$\int e^{x} d x = e^{x} + C$
,
$\int sin x d x = - cos x + C$
.
Linearity:
$\int (a f + b g) d x = a \int f d x + b \int g d x$
.

Power and reciprocal rules

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, \int \frac{d x}{x} = ln ∣ x ∣ + C

Substitution and Partial Fractions

Definition

Substitution

Set $u = g (x)$ so that $d u = g^{'} (x) d x$ , converting a composite integrand into a standard form: $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ .

Look for a function whose derivative is present: the signature of substitution.
Partial fractions break rational expressions into simpler pieces, e.g.
$\frac{1}{( x - 1 ) ( x - 2 )} = \frac{A}{x - 1} + \frac{B}{x - 2}$
.
Use substitution or completing the square when the denominator is quadratic.

Integration by substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u, u = g (x)

Integration by Parts

Integration by parts

\int u d v = uv - \int v d u

Choose u with the ILATE priority: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
Classic uses:
$\int x e^{x} d x$
,
$\int ln x d x$
,
$\int e^{x} sin x d x$
(repeat by parts for cyclic integrals).
For ln x, take u = ln x and dv = dx.

ILATE decides u

Let u be the function that comes first in ILATE. Getting the order wrong is the most common error in integration by parts.

Definite Integrals and the Fundamental Theorem

Definition

Definite integral

\int_{a}^{b} f (x) d x = F (b) - F (a)

The definite integral from a to b equals the change in the antiderivative; geometrically it is the signed area under the curve.

Additivity:
$\int_{a}^{b} + \int_{b}^{c} = \int_{a}^{c}$
.
Symmetric-limit properties:
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$
,
$\int_{- a}^{a} f = 2 \int_{0}^{a} f$
for even f, and 0 for odd f.
King's property and
$\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$
simplify symmetric integrands.

Fundamental theorem and the symmetric property

\int_{a}^{b} f (x) d x = F (b) - F (a), \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

Solved Examples

Example: Evaluate $\int_{0}^{1} x e^{x} d x$ .

Solution: By parts with $u = x$ , $d v = e^{x} d x$ : $\int x e^{x} d x = x e^{x} - \int e^{x} d x = e^{x} (x - 1) + C$ . So $\int_{0}^{1} x e^{x} d x = [e^{x} (x - 1)]_{0}^{1} = 0 - (- 1) = 1$ .

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part II, Chapter 7 (Integrals). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q123)

1) Find an antiderivative (or integral) of the following functions by the method of inspection: (i) ; (ii) ; (iii) ; (iv) ; (v)
2) Find the following integrals: (i) ; (ii)
3) Evaluate: (i) ; (ii) ; (iii)
4) Evaluate: (i) ; (ii)
5) Evaluate: (i) ; (ii)
6) Evaluate: (i) ; (ii) ; (iii)
7) Evaluate:
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50) Integrate the rational functions:
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123) Show that:

Answers at a glance

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CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 12 Mathematics board paper: Section A (18 MCQs × 1M), Section B (5 × 2M), Section C (5 × 3M), Section D (3 × 4M) and Section E (case-study based). Integrals yields 1-mark MCQs on standard integrals, 2/3-mark questions using substitution, integration by parts and partial fractions, and the core 4-mark definite-integrals evaluations with properties like symmetry.

MCQ Questions

The integral of 1/x with respect to x is

∫ cos x dx equals

∫ eˣ dx from 0 to 1 equals

The property ∫₍₀₎ₐ f(x) dx = ∫₍₀₎ₐ f(a − x) dx is used to simplify integrals with

1 Mark Questions

1
Write ∫ xⁿ dx for n ≠ −1. (Ans: xⁿ⁺¹/(n + 1) + C)
2
Evaluate ∫₀¹ x dx. (Ans: 1/2)
3
State the rule for integration by parts for ∫ u dv. (Ans: u∫dv − ∫(du/dx)(∫dv)dx)
4
Evaluate ∫₀¹ 1 dx. (Ans: 1)

2 Mark Questions

1
Find ∫ x/(x² + 1) dx by substitution. (Ans: ½ ln(x² + 1) + C)
2
Evaluate ∫ tan x dx. (Ans: −ln|cos x| + C)
3
Find ∫ x eˣ dx using integration by parts. (Ans: x eˣ − eˣ + C)

3 Mark Questions

1
Find ∫ x sin x dx. (Ans: −x cos x + sin x + C)
2
Evaluate ∫ dx/(x² − 4) using partial fractions. (Ans: (1/4) ln|(x − 2)/(x + 2)| + C)
3
Evaluate ∫₀^π sin x dx and use the symmetry property ∫₀^π f(x)dx = 2∫₍₀₎^(π/2) to confirm. (Ans: 2)

4 Mark Questions

1
Evaluate ∫₀^(π/2) ln(sin x) dx using the substitution x → π/2 − x if needed, and finally using standard results, showing the property used. (Ans: −(π/2) ln 2)
2
Evaluate ∫₀^(π/2) x/(1 + cos x) dx or, with x + y limits, ∫₀^π x sin x/(1 + cos²x) dx using the property ∫₀^a f(x)dx = ∫₀^a f(a − x)dx. (Ans: second integral = π²/4)

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Integrals — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

No Signup Required

Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

Integrals Assignment 1 — Topic-wise Questions

Covers all four topics: indefinite integrals and standard forms; substitution and partial fractions; integration by parts; and definite integrals and the fundamental theorem.

integrals-class-12-notes-assignment-1.pdf

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Integrals Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (standard integrals, substitution, partial fractions, ILATE-based integration by parts, definite integrals and properties) with full worked answers. Includes an answer key.

integrals-class-12-notes-assignment-2.pdf

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Integrals Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

integrals-class-12-notes-test-1-cbse.pdf

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Integrals Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

integrals-class-12-notes-test-2-cbse.pdf

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More Resources

More PDFs for this chapter

Assignments, practice tests and board-pattern papers freshly uploaded for this chapter — download and practise anytime.

Application of Integrals Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

application-of-integrals-class-12-notes-test-2-cbse.pdf

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Application of Integrals Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

application-of-integrals-class-12-notes-test-1-cbse.pdf

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Application of Integrals Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (areas under curves, regions between curves and lines, circles and ellipses, y-integration) with full worked answers. Includes an answer key.

application-of-integrals-class-12-notes-assignment-2.pdf

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Application of Integrals Assignment 1 — Topic-wise Questions

Covers all four topics: area under a curve; area between two curves; standard bounded regions; and choosing the variable.

application-of-integrals-class-12-notes-assignment-1.pdf

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Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Power rule

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

Integration by parts

\int u d v = uv - \int v d u

Fundamental theorem

\int_{a}^{b} f (x) d x = F (b) - F (a)

Symmetric property

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

Linearity

\int (a f + b g) d x = a \int f d x + b \int g d x

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

Always add the constant C for indefinite integrals.
ILATE order for integration by parts: Inverse, Log, Algebraic, Trig, Exponential.
Use ∫₀ᵃ f = ∫₀ᵃ f(a−x) when integrands are symmetric.
∫₋ₐᵃ of an even function = 2∫₀ᵃ; an odd function gives 0.
Substitution works when the derivative of the inside function is present.
n ≠ −1 in the power rule; use ln|x| for 1/x.

FAQ

Frequently asked questions

Why do we add a constant of integration?

Because differentiating F(x) + C removes any constant. The derivative of x² + 5 and x² − 9 are both 2x, so the antiderivative is a whole family F(x) + C.

What is the difference between definite and indefinite integrals?

An indefinite integral is a family of antiderivatives with a constant C; a definite integral is a number, F(b) − F(a), equal to the signed area under the curve.

How do I choose u in integration by parts?

Use the ILATE order — Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential — and take u as the function appearing first.

What is the fundamental theorem of calculus?

It connects differentiation and integration: the definite integral of f from a to b equals the change in any antiderivative F, i.e. ∫ₐᵇ f(x)dx = F(b) − F(a).

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