Integrals Class 12 NCERT Solutions Chapter 7 Ex 7.6

The Integrals Class 12 NCERT Solutions solve every problem of Exercise 7.6 in Class 12 Mathematics Chapter 7 Integrals. The working in the the PDF follows the order taught in the NCERT textbook, so the student can move directly between the solutions PDF page and the corresponding solution without re-reading the theory.

• CBSE Weightage: Exercise 7.6 problems feed the 3-mark and 5-mark slots inside the Integrals 9-11 mark block.
• JEE Main Weightage: Integration by parts contributes roughly 3% of Calculus questions; recurrent in JEE Advanced as well.
• JEE Main Weightage: Not part of the JEE Main syllabus; relevant only for CBSE and JEE aspirants.

Every solved sum in the this chapter opens by labelling u and dv using the ILATE mnemonic (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) so the integration by parts step is never a guess. Special cases like ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C and the repeated-integration trick for ∫ e^ax sin bx dx get full working.

The Collegedunia editorial team has cross-checked every answer against the official NCERT key and the 2026-27 revised textbook. Each problem cites the relevant Section 7.6 formula number so cross-referencing is one-step.

How the Integrals Class 12 NCERT Solutions on the Integrals Class 12 NCERT Solutions Help You

These notes address this in the same order as the NCERT textbook.

Integration by parts is the technique where students lose marks not in the algebra but in choosing the wrong u. The Collegedunia solutions name the ILATE order, write the choice of u and dv on its own line, then apply the formula.

• ILATE labelling at the top of every solve, e.g. "Let u = x (Algebraic), dv = sin x dx (Trigonometric)".
• Standard table giving u, du, dv, v before any substitution, so the uv - ∫ v du plug-in is visible.
• Repeated parts applied without skipping the intermediate step on Q19 (∫ e^x sec x (1 + tan x) dx) and Q22 (∫ tan^-1 x dx).
• Expert's Solution after each main solve where the ∫ e^x [f(x) + f'(x)] dx shortcut would save 2-3 steps.

Integrals Ex 7 6 Video Walkthrough

Source: Magnet Brains on YouTube

What Exercise 7.6 Covers: The Integration by Parts Problem Map

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

Exercise 7.6 problems split neatly by which two functions appear in the product. The 24 sums work through algebraic-trigonometric products, algebraic-exponential products, logarithmic single-function integrals, and inverse-trigonometric single-function integrals.

Three problems (Q3, Q19, Q22) are repeat favourites in CBSE boards. Q22 appeared in the CBSE 2024 board paper as a 5-mark question on ∫ tan^-1 x dx .

Integration by Parts and the ILATE Rule

The the resource address this in the same order as the NCERT textbook.

The integration by parts formula ∫ u dv = uv - ∫ v du only helps when the new integral ∫ v du is simpler than the original. ILATE tells you which factor to call u: pick whichever appears earlier in Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

Two special results worth memorising before solving Exercise 7.6:

• ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C - directly applies to Q19, Q20, Q21.
• For ∫ log x dx , treat the integrand as log x · 1 and pick u = log x, dv = dx. Answer: x log x - x + C .

Class 12 Maths Exercise 7.6 Step-by-Step Approach

Every product-integral problem in this exercise follows the same six-step routine.

1. Identify both factors in the integrand and label each with one ILATE letter.
2. Pick u as the factor whose ILATE letter appears earlier; the remaining factor (together with dx) is dv.
3. Compute du by differentiating u, and compute v by integrating dv (drop the constant of integration here).
4. Apply the formula ∫ u dv = uv - ∫ v du . Write the right-hand side explicitly.
5. Evaluate ∫ v du . If it is still a product, apply integration by parts a second time (with the same ILATE check).
6. Add the constant of integration + C at the very end. Verify by differentiating the answer.

Picking the wrong u is the most common cause of an integration by parts loop that goes nowhere; the ILATE check guards against it.

Class 12th Maths Chapter 7 Exercise 7.6: Solved-Example Highlights

the chapter notes includes full working for every sum. Two highlights below show the typical structure.

Exam Relevance of Class 12 Maths Chapter 7 Exercise 7.6

Exercise 7.6 style sums appear as 3-mark or 5-mark questions in most CBSE board papers. The table below maps the last five sittings.

Across the last five sittings, Exercise 7.6 style sums have carried 3 to 5 marks every single year, making this the most board-relevant exercise inside Chapter 7.

Common Mistakes Students Make in Exercise 7.6

The the PDF are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.

• Forgetting that integration by parts may need to be applied twice on the same problem, e.g. ∫ x² e^x dx and ∫ e^x sin x dx .
• Dropping the negative sign when v = -cos x is substituted into uv - ∫ v du ; the double-negative simplifies to a plus inside the remaining integral.
• Missing the ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C pattern on Q19-Q21 and instead solving the full integration by parts twice.
• Including the constant of integration when computing v from dv; the constant only appears in the final answer.

Integration by Parts Formula Recall for Exercise 7.6

Here are the five most-used formulae for Exercise 7.6, drawn from the Collegedunia formula sheet for Chapter 7.

Full learn sheet: Class 12 Maths Chapter 7 Integrals Formula Sheet.

Related Resources for Class 12 Maths Chapter 7 Integrals

• Formula Sheet for Class 12 Maths Chapter 7 Integrals
• NCERT Notes for Class 12 Maths Chapter 7 Integrals
• NCERT Book PDF for Class 12 Maths Chapter 7 Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Mathematics: All Chapters

Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.

NCERT Solutions for Class 12 Maths Chapter 7: All Exercises

The table below lists every exercise of Class 12 Maths Chapter 7 Integrals currently published on Collegedunia, mapped to the 2026-27 NCERT.

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

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.

All NCERT Solutions for Integrals Ex 7.6 with Step-by-Step Working

Every NCERT textbook question for Class 12 Mathematics Chapter 7 Integrals Ex 7.6 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.

Questions

Class 12 Mathematics Revision Strategy and Exam Practice Routines

Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:

• Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
• Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
• Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
• Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
• Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.

Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.

Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:

• The one-page formula recall sheet built earlier in revision.
• A single full-length mock paper at exam timing.
• Avoid learning any new material the night before — sleep matters more.

Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.

Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.