NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.7 cover all 11 questions on integrals of special types. Collegedunia's subject experts solve every question with the exact formula, the working, and the final answer. The free solutions PDF for Exercise 7.7 is available to download on this page.
- CBSE Weightage: Integrals carries 8-10 marks in the Class 12 Maths board paper
- JEE Main Weightage: Integral Calculus contributes roughly 8-10% of the Mathematics section
- Exercise 7.7 Question Count: 11 questions
Here is what Exercise 7.7 tests, plus the exact formulae to use.
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About Class 12 Maths Chapter 7 Integrals Exercise 7.7
Exercise 7.7 of Chapter 7 Integrals focuses on three standard integrals of special types involving square roots of quadratic expressions. Students apply the direct formulae for ∫√(a² - x²) dx, ∫√(x² - a²) dx, and ∫√(x² + a²) dx, and learn how to reduce more complex integrands to these forms by completing the square.
This page solves all 11 questions of the exercise in the same order as the NCERT textbook, so students can move directly from the question number to the matching solution.
Integrals Exercise 7.7 Solved Step by Step (Video)
Source: Magnet Brains on YouTube
Key Formulae Used in Exercise 7.7
These are the three standard formulae used across Exercise 7.7:
- ∫√(a² - x²) dx = (x/2)√(a² - x²) + (a²/2) sin⁻¹(x/a) + C
- ∫√(x² - a²) dx = (x/2)√(x² - a²) - (a²/2) log|x + √(x² - a²)| + C
- ∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) log|x + √(x² + a²)| + C
Sample Solved Question from Exercise 7.7
This example shows the working style used in every solution on this page.
Question: Evaluate ∫√(x² + 4) dx.
Solution: This matches the standard form ∫√(x² + a²) dx with a = 2.
∫√(x² + 4) dx = (x/2)√(x² + 4) + (4/2) log|x + √(x² + 4)| + C
∫√(x² + 4) dx = (x/2)√(x² + 4) + 2 log|x + √(x² + 4)| + C
Common Mistakes in Exercise 7.7
Most marks lost in Exercise 7.7 come from small slips, not wrong methods.
- Mixing up the sign under the root: ∫√(a² - x²) dx, ∫√(x² - a²) dx, and ∫√(x² + a²) dx each need a different formula. Check the sign before picking one.
- Forgetting the constant of integration C at the end of the answer.
- Completing the square incorrectly before matching a quadratic expression to one of the three standard forms.
- Dropping the modulus sign in the log term for the x² - a² and x² + a² formulas.
How Collegedunia's NCERT Solutions Help
The Collegedunia NCERT Solutions for Exercise 7.7 give every problem a full solved method, name the standard form used, and show each substitution step clearly. Every solution follows the format CBSE evaluators expect, so students score full marks on integration questions in the board paper.
Other Resources for Class 12 Maths Chapter 7 Integrals
Pair the Exercise 7.7 solutions with the rest of the Chapter 7 resource library for complete preparation.
| Resource | What it covers |
|---|---|
| Chapter 7 Full Solutions | Every exercise of the chapter, solved in one place |
| Chapter 7 Notes | Theory, definitions, solved examples |
| Chapter 7 Formula Sheet | All formulae on one page |
| Chapter 7 Handwritten Notes | Scanned-style revision notes |
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.
| Exercise | Topic Tested |
|---|---|
| Exercise 7.1 | Indefinite integrals; standard formulas |
| Exercise 7.2 | Integration by substitution |
| Exercise 7.3 | Integration using trigonometric identities |
| Exercise 7.4 | Integrals of special functions |
| Exercise 7.5 | Integration by partial fractions |
| Exercise 7.6 | Integration by parts |
| Exercise 7.7 | Integrals of special types |
| Exercise 7.8 | Definite integrals; fundamental theorem of calculus |
| Exercise 7.9 | Evaluation of definite integrals by substitution |
| Exercise 7.10 | Properties of definite integrals |
| Miscellaneous Exercise | Mixed indefinite and definite integration problems |
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.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
All NCERT Solutions for Integrals Ex 7.7 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 7 Integrals Ex 7.7 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
Integrate √4 - x2.
Integrate √1 - 4 x2.
Integrate √x2 + 4 x + 6.
Integrate √x2 + 4 x + 1.
Integrate √1 - 4 x - x2.
Integrate √x2 + 4 x - 5.
Integrate √1 + 3 x - x2.
Integrate √x2 + 3 x.
Integrate √1 + x29.
∫ √1 + x2 dx is equal to
(A) x2√1 + x2 + 12log(x + √1 + x2) + C
(B) 23(1 + x2)3/2 + C
(C) 23 x (1 + x2)3/2 + C
(D) x22√1 + x2 + 12 x2 log|x + √1 + x2| + C
∫ √x2 - 8 x + 7 dx is equal to
(A) 12(x - 4)√x2 - 8 x + 7 + 9log|x - 4 + √x2 - 8 x + 7| + C
(B) 12(x + 4)√x2 - 8 x + 7 + 9log|x + 4 + √x2 - 8 x + 7| + C
(C) 12(x - 4)√x2 - 8 x + 7 - 3√2log|x - 4 + √x2 - 8 x + 7| + C
(D) 12(x - 4)√x2 - 8 x + 7 - 92log|x - 4 + √x2 - 8 x + 7| + C
Student Feedback - Integrals Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Integrals Class 12 NCERT Solutions - Frequently Asked Questions
Ques. What does Exercise 7.7 of Chapter 7 Integrals cover?
Ans. Exercise 7.7 covers integrals of special types involving √(a² - x²), √(x² - a²), and √(x² + a²), plus integrals reducible to these by completing the square.
Ques. How many questions are in Class 12 Maths Ex 7.7?
Ans. Exercise 7.7 contains 11 problems applying the three standard integrals of special types.
Ques. Are these solutions based on the 2026-27 NCERT syllabus?
Ans. Yes, all solutions follow the latest 2026-27 NCERT textbook and CBSE marking scheme.
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