Integrals Class 12 Notes - Class 12 Maths Ch 7

Get the Integrals Class 12 Notes below as a free PDF for Class 12 Maths Chapter 7 Integrals. The notes follow the 2026-27 NCERT structure and reproduce the standard solved examples. The free download sits right here on this page.

• CBSE Boards: Integrals carries 6-8 marks, and with Application of Integrals the calculus unit totals 9-12 marks.
• JEE Main: about 2-3 questions, with definite integrals and integration by parts as repeat favourites.
• CUET UG (Maths): 3-5 MCQs on standard forms and property P4.

What makes these Integrals Class 12 Notes effective for revision

These notes are built for speed, technique-selection and accuracy. Most students know the formulas, but lose marks by picking the wrong method.

• Every section maps to the NCERT flow (7.1 to 7.7) so you revise in order.
• Each technique carries a "when to use it" cue, and the six special forms sit in one block for quick recall.

Integrals Class 12 Chapter 7 Explained Step by Step (Video)

Source: Magnet Brains on YouTube

Class 12 Maths Chapter 7 Integrals: topic-by-topic breakdown

Integration as an anti-derivative

Integration reverses differentiation. If ddx F(x) = f(x) , then ∫ f(x) dx = F(x) + C .

Memorise the basics such as ∫ xⁿ dx = xⁿ⁺¹n+1 + C (n ≠ -1) and ∫ 1x dx = ln|x| + C . Never drop the modulus in the log; examiners deduct half a mark for it.

Integration by substitution

Use substitution when one part of the integrand is the derivative of another. A key result is ∫ f'(x)f(x) dx = ln|f(x)| + C .

Watch Out: in a definite integral, change the limits when you change the variable. Forgetting this is a common 2-mark loss.

Integration using trigonometric identities

Powers of sine and cosine rarely integrate directly, so rewrite them first using sin² x = 1 - cos 2x2 and product-to-sum identities.

Integrals of special functions (the six standard forms)

NCERT lists six template integrals. Complete the square whenever the denominator is quadratic.

$$ \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C $$

$$ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C $$

$$ \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\frac{x}{a} + C $$

$$ \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C $$

$$ \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left|x + \sqrt{x^2 + a^2}\right| + C $$

$$ \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left|x + \sqrt{x^2 - a^2}\right| + C $$

Integration by partial fractions

For a proper rational function, decompose before integrating. NCERT covers distinct linear, repeated linear, distinct quadratic and repeated quadratic factors.

Board Tip: if the fraction is improper, divide first. Examiners trap students on this.

Integration by parts and the LIATE rule

In by-parts, choose u by the LIATE order: Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential. A must-know result is ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C .

Pro-Tip: for ∫ e^x sin x dx , apply by-parts twice and rearrange to solve algebraically.

Definite integrals and properties

By the Fundamental Theorem, if F'(x) = f(x) then _a^b f(x) dx = F(b) - F(a) . The most-tested property is P4: ₀^a f(x) dx = ₀^a f(a-x) dx .

Exam Strategy: when limits are 0 to π/2 with sin or cos, try P4 first; it usually collapses the question to one line.

Definite integral as the limit of a sum (JEE)

The Riemann-sum form is non-evaluative for boards but appears in JEE Main. Practise converting a limit-of-sum back into a definite integral by spotting h = (b-a)/n.

NCERT Class 12 Maths Chapter 7 Integrals: Important Topics

This table is a quick pre-exam reference to the topics that matter most.

Integrals: Most Repeated Questions in Board Exams

These question types resurface almost every session.

• Evaluate ₀^π/2 sin xsin x + cos x dx using P4; the answer is π/4 (5-mark, 2018/2020/2023).
• Evaluate ∫ x(x-1)(x-2) dx by partial fractions (3-mark, 2017/2021).

Integrals in recent CBSE papers: a 5-year analysis

Across 2020 to 2024, the 5-mark long-answer almost always tests property P4, by-parts appears as a 3-marker, and the MCQ tests a standard formula.

Weightage of Integrals in JEE Main and CUET

Common mistakes students make in Class 12 Integrals

Even strong students drop marks on recurring traps. Watch for these:

• Sign errors in substitution: when t = cos x, dt = -sin x dx. Write dt out fully.
• Not changing limits in a definite integral after substituting.
• Wrong partial-fraction setup for a repeated linear factor (you need both terms).
• Forgetting the +C in indefinite integrals, or picking the wrong u in by-parts.

Integrals Class 12 Notes: Quick Summary

• The notes cover every section of Chapter 7 Integrals, aligned to NCERT 2026-27.
• They include definitions, solved examples and a formula recap, and the free PDF follows the NCERT textbook notation.

Exercise-wise Breakdown of the Integrals Chapter

The chapter splits into 10 exercises plus a Miscellaneous Exercise, each linked to its worked solutions.

PDF Download Formats for the Integrals Chapter

The Integrals Class 12 PDF comes in three formats.

All formats follow the printed NCERT 2026-27 notation, and a Hindi-medium edition is available.

Important Questions and Previous Year Trends for the Integrals Chapter

Three question templates account for most of the marks this chapter contributes.

How the Integrals Notes Pair with NCERT Solutions and the Formula Sheet

How to Use the Integrals Notes Page Most Effectively

Split your revision across three sittings for the best result.

Other Resources for Class 12 Maths Chapter 7 Integrals