Integrals Class 12 Notes - Class 12 Maths Ch 7

Get the Integrals Class 12 Notes below as a free PDF reference for Class 12 Mathematics Chapter 7 Integrals. The Integrals Class 12 Notes follow the 2026-27 NCERT structure and reproduce the standard solved examples taught in class. The notes are written in simple, formal mathematical prose.

• CBSE Boards: Integrals carries 6–8 marks on its own; combined with Application of Integrals, the calculus unit totals 9–12 marks every year.
• JEE Main: Around 2–3 questions (~5% of the Maths section), with definite integrals and integration by parts being repeat favourites.
• CUET UG (Maths/Applied Maths): Expect 3–5 MCQs from indefinite and definite integration combined, frequently testing standard forms and property P4.

If you want to revise the entire Class 12 Maths Integrals chapter in a single sitting before a board paper, the Integrals Class 12 Notes below cover every standard form, special integral and definite-integral property you are likely to be tested on - with solved examples that show which technique to reach for first.

These Integrals Class 12 Notes have been built by Class 12 Maths specialists, mapped to the 2025–26 CBSE syllabus and refined against the last five years of board and JEE Main papers, so what you revise is exactly what tends to appear in the exam.

What makes these Class 12 the Integrals Class 12 Notes effective for revision

We have structured this set of notes around three goals: speed, technique-selection and accuracy. Most students know the formulas - the marks are lost in picking the wrong method or forgetting a sign.

• Every section is mapped to the NCERT flow (sections 7.1 to 7.7) so you can revise alongside the Integrals Class 12 Notes without losing your place.
• Each integration technique is paired with a "when to use it" cue so you stop guessing and start recognising the question type within 10 seconds.
• Special integral forms (the six radical/quadratic templates) are presented as a single comparison block so you can spot the right substitution at a glance.
• All formulas use MathJax for crisp rendering on mobile, and the most common board-exam traps are flagged with watch-out callouts.

Integrals Video Walkthrough

Source: Magnet Brains on YouTube

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

Below is the walkthrough - every concept from indefinite integration to limit-of-a-sum, in the order the NCERT chapter introduces them.

Integration as an anti-derivative

Integration is the inverse of differentiation. If ddx F(x) = f(x) , then ∫ f(x) dx = F(x) + C , where C is the arbitrary constant of integration. The family of all anti-derivatives of f is called the indefinite integral of f .

Standard results you must memorise cold include ∫ xⁿ dx = xⁿ⁺¹n+1 + C (for n ≠ -1 ), ∫ 1x dx = ln|x| + C , ∫ e^x dx = e^x + C , and the six trigonometric anti-derivatives.

Memory Hook: Differentiation knocks the power down; integration boosts it back up and divides. Never drop the modulus in ln|x| - examiners deduct half a mark for it.

Integration by substitution - when the integrand hides a derivative

Use substitution when one part of the integrand is (or differs by a constant from) the derivative of another part. Set t = g(x) , so dt = g'(x) dx , and the integral collapses into a standard form.

Classic targets: ∫ tan x dx = ln|sec x| + C , ∫ f'(x)f(x) dx = ln|f(x)| + C , and any integrand involving sinⁿ x cos x or e^x2 · x .

Watch Out: When you change the variable in a definite integral, change the limits too. Forgetting this is the single most common 2-mark loss in the Integrals Class 12 Notes.

Integration using trigonometric identities

Powers and products of sines and cosines rarely integrate directly - you first rewrite them using identities.

Use sin² x = 1 - cos 2x2 , cos² x = 1 + cos 2x2 , and product-to-sum identities like 2sin A cos B = sin(A+B) + sin(A-B) . Tangent-secant integrals such as ∫ sec³ x dx need a by-parts trick on top of identities.

Integrals of special functions (the six standard forms)

NCERT lists six template integrals that recur across boards and JEE. Memorise these as a block:

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

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

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

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

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

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

Whenever the denominator is quadratic, complete the square first so the integrand fits one of these six shapes.

Integration by partial fractions - for rational functions

When the integrand is a proper rational function P(x)Q(x) , decompose it before integrating. NCERT covers four cases: distinct linear factors, repeated linear factors, distinct quadratic factors and repeated quadratic factors. For example, 1(x-1)(x-2) = -1x-1 + 1x-2 , giving ln|x-2x-1| + C .

Board Tip: If the rational function is improper (degree of numerator ≥ degree of denominator), divide first to split it into polynomial + proper fraction. Examiners regularly trap students on this in 3-mark questions.

Integration by parts and the LIATE rule

The by-parts formula is ∫ u · v dx = u ∫ v dx - ∫ ( dudx ∫ v dx ) dx . The trick is choosing which factor is "u" (to be differentiated) and which is "v" (to be integrated). Use the LIATE priority order to decide u:

• L - Logarithmic (e.g., ln x )
• I - Inverse trigonometric (e.g., tan^-1 x )
• A - Algebraic (e.g., x² )
• T - Trigonometric (e.g., sin x )
• E - Exponential (e.g., e^x )

Whichever class appears first in LIATE becomes u. The two NCERT "by-parts results" worth memorising are ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C and ∫ √a² - x² dx = x2√a² - x² + a²2sin^-1xa + C .

Pro-Tip: If you cannot decide using LIATE - for instance ∫ e^x sin x dx - apply by-parts twice and rearrange. The original integral will reappear and you solve algebraically.

Definite integrals and the Fundamental Theorem of Calculus

The definite integral _a^b f(x) dx measures the net signed area between the curve and the x-axis from a to b. The Second Fundamental Theorem of Calculus links it to anti-derivatives:

if F'(x) = f(x) , then _a^b f(x) dx = F(b) - F(a) . The First FTC states that ddx _a^x f(t) dt = f(x) . These two statements together let you evaluate definite integrals without going back to limits-of-sums.

Properties of definite integrals (especially P4)

NCERT lists seven properties (P0 to P7) of definite integrals. The most frequently tested is P4: ₀^a f(x) dx = ₀^a f(a-x) dx . It appears in nearly every CBSE 5-marker for the Integrals Class 12 Notes. Adding the original integral to its P4-transformed version often simplifies trig denominators dramatically. Other heavy-use properties include:

• P1: _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
• P6: ₀^2a f(x) dx = 2₀^a f(x) dx if f(2a-x) = f(x) , else 0.
• P7: _-a^a f(x) dx = 2₀^a f(x) dx if f is even, 0 if f is odd.

Exam Strategy: The moment you see limits 0 to π/2 or 0 to π with sin/cos in the integrand, try P4 first - it usually collapses the question into one line.

Definite integral as the limit of a sum (JEE extension)

The Riemann-sum definition _a^b f(x) dx = _{n → ∞} b-an _r=1ⁿ f(a + rb-an) is non-evaluative for boards but appears in JEE Main as a 4-mark question almost every year. Practise converting a given limit-of-sum back into a definite integral by identifying h = (b-a)/n and recognising f(x).

NCERT Class 12 Maths Chapter 7 Integrals: Important Topics

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Integrals: Most Repeated Questions in Board Exams

By analysing the previous year's trends, we have identified the question patterns that resurface almost every session. Practise these before you walk into the board exam.

Ques. Evaluate ∫ 1√x² + 2x + 5 dx (2019, 2022, 2024)

[3-Mark Question]

Complete the square: x² + 2x + 5 = (x+1)² + 2² . It now fits the standard form ∫ dx√x² + a² , giving ln|(x+1) + √(x+1)² + 4| + C .

Ques. Evaluate ₀^π/2 sin xsin x + cos x dx (2018, 2020, 2023)

[5-Mark Question]

This is the Integrals Class 12 Notes P4 application. Let I be the given integral and apply x → π/2 - x to get a second expression. Adding the two integrals yields 2I = ₀^π/2 1 dx = π/2 , so I = π/4.

Ques. Evaluate ∫ x(x-1)(x-2) dx (2017, 2021)

[3-Mark Question]

Partial fractions: x(x-1)(x-2) = -1x-1 + 2x-2 . Integrating term-by-term gives -ln|x-1| + 2ln|x-2| + C , or ln|(x-2)²x-1| + C .

Ques. Find ∫ x² · e^x dx (2019, 2023)

[3-Mark Question]

By LIATE, u = x² (Algebraic) and v = eˣ (Exponential). Two applications of by-parts give e^x(x² - 2x + 2) + C . Verify by differentiating.

Ques. Evaluate ₀^π x sin x1 + cos² x dx (2020, 2024)

[5-Mark Question]

Apply P4 with a = π. After substitution and addition, the integral reduces to π2₀^π sin x1 + cos² x dx . Now put t = cos x, dt = −sin x dx, and the answer comes out to π²/4.

Integrals in recent CBSE papers - a 5-year analysis

Across the 2020–2024 CBSE Class 12 Maths question papers, Chapter 7 has been one of the steadiest contributors, with a predictable mix of MCQ, short-answer and long-answer slots.

• The 5-mark long-answer question almost always tests a property of definite integrals - P4 specifically appears in roughly 4 out of every 5 papers.
• Integration by parts shows up either as a 3-marker ( ∫ x · e^x dx family) or folded into a 5-marker with by-parts twice.
• Partial fractions appear consistently as a 3-marker - typically with one repeated linear factor or one irreducible quadratic.
• The MCQ slot usually tests a standard formula directly (e.g., ∫ sec² x dx ) or asks for the value of a small definite integral.

Source:

• Class 12 Maths Chapter 7 NCERT Solutions
• CBSE Class 12 Maths Hub - Syllabus, Papers, Resources

Weightage of Integrals in JEE Main, JEE Main & CUET

Beyond boards, Integrals is one of the highest-yield chapters in competitive Maths papers. JEE Main typically asks 2–3 calculus questions, and at least one is reliably from this chapter - often combining definite integrals with limit-of-a-sum or by-parts.

• For JEE Main, focus extra on definite-integral properties, limit-of-a-sum and parametric definite integrals (e.g., differentiation under the integral sign as a bonus topic).
• For CUET, lean heavily on standard formulas and partial fractions - the MCQ format rewards quick pattern recognition.

Common mistakes students make in Class 12 Integrals

Even strong students drop marks on this chapter because of recurring traps. Here are the most common ones to watch for:

• Sign errors during substitution: When you set t = cos x, dt = −sin x dx - the negative sign is missed in roughly half of all submitted scripts. Always write down dt explicitly.
• Forgetting to change limits in definite integrals after substitution: If you substitute and don't switch to the new limits, you must back-substitute before plugging in - most students do neither and lose 2 marks instantly.
• Wrong partial-fraction setup: For a repeated linear factor (x-a)² , the decomposition needs both Ax-a and B(x-a)² . Skipping the second term gives the wrong answer.
• Picking the wrong "u" in by-parts: Students often choose the function that's easier to integrate as u instead of following LIATE. The result balloons in complexity and rarely closes out.
• Treating an improper rational function like a proper one: If degree(numerator) ≥ degree(denominator), divide first. Jumping straight to partial fractions on x³x²+1 is a textbook 3-mark loss.
• Forgetting the +C in indefinite integrals: A small slip examiners actively dock half a mark for. Make it a reflex.
• Not spotting when P4 applies: Whenever you see ₀^a with a symmetric integrand - pause, write down P4 and test it before grinding through manually.

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

Integrals Class 12 Notes - Quick Summary

• The Integrals Class 12 Notes cover every section of Class 12 Mathematics Chapter 7 Integrals, aligned to the 2026-27 NCERT print.
• The Integrals Class 12 Notes include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
• The Integrals Class 12 Notes are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.

Exercise-wise Breakdown of the Integrals Chapter

The Integrals chapter splits into 10 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.

PDF Download Formats and Languages for the Integrals Chapter

The Integrals Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:

The integrals class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:

• NCERT-faithful: Every definition, theorem and exercise on the integrals class 12 ncert pdf matches the printed textbook line for line.
• Hindi-medium edition: The integrals class 12 pdf is also available in Hindi - same page numbering, same equation labels.
• Formula PDF separate: The integrals class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
• Solutions PDF separate: The integrals class 12 solutions pdf gives every NCERT exercise worked out step by step.
• State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.

Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the integrals class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.

Important Questions and Previous Year Trends for the Integrals Chapter

The most repeated question patterns in CBSE Class 12 Maths for the Integrals chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:

Walking through one example of each template before the exam covers most of the predictable integrals class 12 important questions you will see on board day.

• integrals class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
• The integrals class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
• For NCERT Exemplar practice, the matching integrals class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
• The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.

Year-wise PYQ Distribution

The table below maps the dominant question type asked from the Integrals chapter across recent CBSE Class 12 Maths boards:

The full integrals class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.

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

The Integrals Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:

Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.

• The integrals class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
• The integrals class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
• The integrals class 12 formulas reference sheet is the same A4 file students sometimes refer to as integrals class 12 all formulas - it lists every identity used in the chapter.
• State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board integrals class 12 textbook PDF all share the same core definitions.
• For class-first search phrasings - class 12 integrals solutions, class 12 integrals ncert solutions, ncert class 12 integrals solutions - the same files cover the request.

Reference Books and State-Board Mapping

Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:

How to Use the Integrals Notes Page Most Effectively

The recommended study plan for the Integrals Class 12 chapter splits across three sittings. The table below outlines what to do in each:

For students preparing for both CBSE board and JEE Main:

• 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
• 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
• The integrals class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
• For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.

This Collegedunia NCERT Class 12 Mathematics page is reviewed against every CBSE board paper release.