NCERT Solutions for Class 12 Maths Chapter 7 (Ex 7.9) Integrals
FAQs on NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9
1. What is the primary topic covered in the NCERT Solutions for Class 12 Maths Chapter 7, Exercise 7.9?
The NCERT Solutions for Class 12 Maths Chapter 7, Exercise 7.9, primarily focus on the Evaluation of Definite Integrals by Substitution. This exercise provides step-by-step methods for solving definite integrals where a direct integration is complex, in line with the CBSE 2025-26 syllabus.
2. How is the substitution method used to solve definite integrals in Exercise 7.9?
The substitution method involves a series of steps to simplify and solve the integral:
- Choose a substitution: Identify a part of the integrand to substitute with a new variable, say u.
- Find the differential: Calculate the differential du in terms of dx.
- Change the limits: Convert the original limits of integration (which are for x) to the corresponding limits for the new variable u.
- Integrate: Solve the simplified integral with respect to u using the new limits.
- Evaluate: Calculate the final value by finding the difference of the function at the upper and lower limits.
3. How many questions are there in Exercise 7.9 of NCERT Class 12 Maths, and what is their main purpose?
Exercise 7.9 contains 10 questions. The main purpose of these questions is to provide students with thorough practice in applying the substitution method to a variety of functions, including trigonometric, algebraic, and inverse trigonometric functions, to build problem-solving proficiency.
4. Why is it essential to change the limits of integration when using the substitution method for definite integrals?
It is essential because the original limits are defined for the variable x. When you change the variable of integration from x to u, the entire integral, including its bounds, must be expressed in terms of u. Applying the old x-limits to the new u-based integral would be mathematically incorrect and would lead to the wrong answer. The new limits correspond to the values of u when x is at its original lower and upper bounds.
5. What is a common mistake students make while solving problems from NCERT Exercise 7.9?
A very common mistake is forgetting to change the limits of integration after substituting the variable. Students often perform the integration with the new variable 'u' but incorrectly apply the original 'x' limits for evaluation. Another frequent error is making a mistake while finding the new differential (e.g., dx in terms of du).
6. How do you solve a problem like Question 7 from Ex 7.9, ∫dx/(x² + 2x + 5) from -1 to 1?
To solve this, you use the method of completing the square and then substitution. The steps are:
1. Rewrite the denominator: x² + 2x + 5 can be written as (x² + 2x + 1) + 4, which simplifies to (x + 1)² + 2².
2. The integral becomes ∫dx/((x + 1)² + 2²).
3. Substitute: Let u = x + 1. Then du = dx.
4. Change limits: When x = -1, u = 0. When x = 1, u = 2.
5. The new integral is ∫du/(u² + 2²) from 0 to 2.
6. This is a standard integral form, which evaluates to (1/2)tan⁻¹(u/2).
7. Apply the new limits [0, 2] to get the final answer, which is π/8.
7. Can all definite integrals in Chapter 7 be solved using the substitution method from Exercise 7.9?
No, the substitution method is not a universal solution. It is effective only when the integrand contains a function and its derivative (or a function that can be simplified via substitution). For other types of integrals, you must use other methods taught in Chapter 7, such as integration by parts, integration using trigonometric identities, or integration by partial fractions.
8. How does mastering the substitution method in Exercise 7.9 help in preparing for board exams?
Mastering the substitution method is crucial as it is a foundational technique frequently tested in the CBSE board exams. It improves problem-solving speed and accuracy. Furthermore, this method is a prerequisite for understanding more advanced topics like finding the area under curves (Chapter 8) and solving differential equations (Chapter 9), which carry significant weightage in exams.

















