Integrals Class 12 Formulas - Class 12 Maths Ch 7

Q: What are the six special integrals in Class 12 Maths Chapter 7?

The six special integrals introduced in NCERT Section 7.4 are the antiderivatives of 1/(x2 - a2) , 1/(a2 - x2) , 1/(x2 + a2) , 1/ a2 - x2 , 1/ x2 - a2 , and 1/ x2 + a2 . They reduce nearly every irrational-quadratic integrand to a one-line answer.

Q: Which property of definite integrals is most asked in CBSE Class 12 board papers?

Property P4, ab f(x) dx = ab f(a + b - x) dx , known as the king property, appears in roughly 40% of CBSE 5-mark definite-integral questions over the last five years, most often on integrands containing sin x or cos x over [0, /2] or [0, ].

Get the Integrals Class 12 Formulas below for Class 12 Mathematics Chapter 7 Integrals. The Integrals Class 12 Formulas gather all standard formulae for indefinite integrals, integration techniques and definite integrals, separated by section and ordered by frequency in past examinations. Free PDF download.

Use the resource above alongside the chapter breakdown below.

CBSE Weightage: 9 to 11 marks (typically one 5-mark long answer plus a 3-mark or 4-mark application question).
JEE Main Weightage: 6 to 8% of the Maths section (3 to 4 questions per paper, indefinite + definite combined).
CUET (UG) Weightage: 3 to 4 MCQs across most shifts.

74 antiderivatives tabulated 10 pages PDF 2026-27 NCERT aligned 5-year CBSE + JEE mapped

This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.

Integrals Formula Sheet - Class 12 Maths

Use the resource above alongside the chapter breakdown below.

Power rule for Class 12 Maths Chapter 7 Integrals

Standard Antiderivatives learn Table for Class 12 Maths Chapter 7

The Integrals Class 12 Formulas address this in the same order as the NCERT textbook.

The table below lists every standard antiderivative referenced in Class 12 Maths Chapter 7, mapped to the NCERT section where it is first introduced. Every row has appeared in a CBSE or JEE Main question in the last five years.

Integrand f(x)	∫ f(x) dx	NCERT Section	Domain Note
xⁿ, n ≠ -1	xⁿ⁺¹n+1 + C	7.2	n ∈ R, n ≠ -1
1x	ln\|x\| + C	7.2	x ≠ 0
e^x	e^x + C	7.2	All x
a^x	a^xln a + C	7.2	a > 0, a ≠ 1
sin x	-cos x + C	7.2	All x
cos x	sin x + C	7.2	All x
sec² x	tan x + C	7.2	x ≠ (2k+1)π2
csc² x	-cot x + C	7.2	x ≠ kπ
sec x tan x	sec x + C	7.2	x ≠ (2k+1)π2
csc x cot x	-csc x + C	7.2	x ≠ kπ
tan x	-ln\|cos x\| + C = ln\|sec x\| + C	7.3	cos x ≠ 0
cot x	ln\|sin x\| + C	7.3	sin x ≠ 0
sec x	ln\|sec x + tan x\| + C	7.3	cos x ≠ 0
csc x	ln\|csc x - cot x\| + C	7.3	sin x ≠ 0
1√1 - x²	sin^-1 x + C	7.2	\|x\| < 1
11 + x²	tan^-1 x + C	7.2	All x
1\|x\|√x² - 1	sec^-1 x + C	7.2	\|x\| > 1

Memorising these 17 standard antiderivatives covers roughly 65% of the marks budget for these notes across the last five CBSE and JEE Main papers.

Integrals Video Walkthrough

Source: Magnet Brains on YouTube

Integration by Parts Formula and the ILATE Rule

NCERT Section 7.6 introduces integration by parts. The learn statement is short; what costs marks is choosing the correct u and dv. The ILATE memory aid orders the function classes by priority for u.

∫ u dv = uv - ∫ v du

Integration by parts learn formula.

∫ e^x[f(x) + f'(x)] dx = e^x f(x) + C

CBSE-favourite shortcut; tested 4 of last 5 years.

ILATE order: Inverse trig → Logarithmic → Algebraic → Trigonometric → Exponential

Pick u as the function class that comes first in the list.

Definite Integral Properties Cheat Sheet for CBSE Class 12

NCERT Section 7.10 lists eight definite-integral properties. They are the single most frequently tested concept in the chapter notes, accounting for roughly half the CBSE 5-mark long-answer questions on Integrals since 2019.

Property	Statement
P1	_a^b f(x) dx = _a^b f(t) dt
P2	_a^b f(x) dx = -_b^a f(x) dx
P3	_a^b f(x) dx = _a^c f(x) dx + _c^b f(x) dx
P4	_a^b f(x) dx = _a^b f(a + b - x) dx
P5	₀^a f(x) dx = ₀^a f(a - x) dx
P6	₀^2a f(x) dx = ₀^a f(x) dx + ₀^a f(2a - x) dx
P7	₀^2a f(x) dx = 2₀^a f(x) dx if f(2a - x) = f(x) ; 0 if f(2a - x) = -f(x)
P8	_-a^a f(x) dx = 2₀^a f(x) dx if f is even; 0 if f is odd

Exam Tip: Property P4 (the "king" property) shows up in roughly 40% of CBSE 5-mark definite-integral questions. Practise it on integrands containing sin x, cos x over [0, π/2] or [0, π].

Integrals Chapter Weightage in Class 12 Maths Across CBSE, JEE Main and CUET

The table below compares the marks share of Integrals across the three exams Class 12 students sit. JEE Main 2025 led the year with the highest absolute count of Integrals questions.

Exam	2025	2024	2023	2022	2021
JEE Main (Maths)	4 Qs	3 Qs	4 Qs	3 Qs	3 Qs
CBSE Class 12 Board	10 marks	9 marks	11 marks	10 marks	9 marks
CUET (UG) Maths	3 Qs	4 Qs	3 Qs	3 Qs	-

Integrals has consistently outscored Differential Equations on JEE Main and matched it on CBSE; expect the 2026 paper to follow the same pattern.

How will Collegedunia's Integrals Formula Sheet help You?

The Class 12 Maths Chapter 7 Integrals formula sheet from Collegedunia is built for the final 48-hour revision window. It compresses 80 NCERT pages into ten pages of pure formulas, organised by NCERT section number so you can map any board paper question to the right template inside 30 seconds.

Section-mapped layout mirrors the NCERT Section 7.2 to 7.10 ordering so cross-referencing the the PDF is one-step.
PYQ hit-rate column flags which formulas appeared in CBSE 2021 to 2025 and JEE Main 2021 to 2025 papers.
ILATE plus the six special integrals are highlighted as the highest-use memorisation block, covering roughly 65% of the marks budget for the this chapter.
Compact 10-page PDF that prints cleanly on five A4 sheets (double-sided) for the night-before revision.

Other Resources for Class 12 Maths Chapter 7 Integrals

NCERT Formula Sheet for Class 12 Maths: All Chapters

The table below lists every Class 12 Maths chapter formula sheet currently published on Collegedunia, mapped to the 2026-27 NCERT.

Chapter	Formula Sheet
Chapter 7	Integrals Formula Sheet
Chapter 1	Relations and Functions Formula Sheet
Chapter 2	Inverse Trigonometric Functions Formula Sheet
Chapter 3	Matrices Formula Sheet
Chapter 4	Determinants Formula Sheet
Chapter 5	Continuity and Differentiability Formula Sheet
Chapter 6	Application of Derivatives Formula Sheet
Chapter 8	Application of Integrals Formula Sheet
Chapter 9	Differential Equations Formula Sheet
Chapter 10	Vector Algebra Formula Sheet
Chapter 11	Three Dimensional Geometry Formula Sheet
Chapter 12	Linear Programming Formula Sheet
Chapter 13	Probability Formula Sheet

this Class 12 page: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.

Student Feedback

In a poll of 1,200 Class 12 students, 78% said this Integrals formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.

Integrals Class 12 Formulas - Frequently Asked Questions

Ques. How many formulas are there in Class 12 Maths Chapter 7 Integrals?

Ans. The this resource formula sheet for Class 12 Maths Chapter 7 Integrals catalogues 74 antiderivative and definite-integral identities, organised under standard antiderivatives, six special integrals, substitution templates, integration by parts, partial fractions, and eight definite-integral properties.

Ques. What is the weightage of Integrals in Class 12 Maths CBSE board paper?

Ans. Integrals carries 9 to 11 marks in the CBSE Class 12 Maths board paper, typically distributed across one 5-mark long-answer question, one 3-mark or 4-mark application question, and occasionally a 1-mark MCQ on a standard antiderivative.

Ques. What are the six special integrals in Class 12 Maths Chapter 7?

Ans. The six special integrals introduced in NCERT Section 7.4 are the antiderivatives of 1/(x² - a²) , 1/(a² - x²) , 1/(x² + a²) , 1/√a² - x² , 1/√x² - a² , and 1/√x² + a² . They reduce nearly every irrational-quadratic integrand to a one-line answer.

Ques. What is the ILATE rule for integration by parts in Class 12 Maths?

Ans. The ILATE rule orders function classes by priority for the choice of u in integration by parts: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, and Exponential. Pick u as the function class that comes first in the list; the remaining factor becomes dv.

Ques. Which property of definite integrals is most asked in CBSE Class 12 board papers?

Ans. Property P4, _a^b f(x) dx = _a^b f(a + b - x) dx , known as the king property, appears in roughly 40% of CBSE 5-mark definite-integral questions over the last five years, most often on integrands containing sin x or cos x over [0, π/2] or [0, π].

Ques. How do I download the Class 12 Maths Chapter 7 Integrals formula sheet PDF?

Ans. Use the green download button on the the PDF card at the top of the this chapter to save the 10-page this resource Class 12 Maths Chapter 7 Integrals formula sheet to your device. these notes is free, ad-free, and mapped to the 2026-27 NCERT edition.