Maths Class 12 Chapter 7 Questions and Answers - Free PDF Download
FAQs on NCERT Solutions For Class 12 Maths Chapter 7 Integrals Exercise 7.1 - 2025-26
1. What is the correct step-by-step method to solve problems in NCERT Class 12 Maths Chapter 7, Integrals?
To solve integration problems from Chapter 7 as per the CBSE pattern, follow this standard methodology:
- Identify the Integrand: First, carefully examine the function to be integrated to determine its type (e.g., algebraic, trigonometric, exponential).
- Choose the Right Method: Select the appropriate integration technique. For basic functions, use standard formulas. For complex functions, use integration by substitution, by parts, or by partial fractions.
- Apply the Method: Execute the chosen method step-by-step, showing all transformations and substitutions clearly.
- Integrate: Perform the integration using the relevant formulas.
- Add the Constant of Integration: For indefinite integrals, always add the constant '+ C' to your final answer to represent the entire family of antiderivatives.
2. What are the fundamental integration formulas I must know for Chapter 7?
Mastering Chapter 7 requires a strong command of the following essential integration formulas from the NCERT syllabus:
- ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
- ∫(1/x) dx = ln|x| + C
- ∫eˣ dx = eˣ + C
- ∫aˣ dx = (aˣ / ln a) + C
- ∫sin(x) dx = -cos(x) + C
- ∫cos(x) dx = sin(x) + C
- ∫sec²(x) dx = tan(x) + C
- ∫csc²(x) dx = -cot(x) + C
- ∫sec(x)tan(x) dx = sec(x) + C
- ∫csc(x)cot(x) dx = -csc(x) + C
3. How do I decide when to use integration by substitution, by parts, or by partial fractions?
Choosing the correct integration technique is crucial. Here’s a simple guide:
- Use Integration by Substitution when the integrand contains a function and its derivative (e.g., ∫f(g(x))g'(x)dx). This method simplifies the integral into a standard form.
- Use Integration by Parts when the integrand is a product of two different types of functions, like logarithmic, inverse, algebraic, trigonometric, or exponential (follow the ILATE rule). It is used for integrals like ∫x cos(x) dx.
- Use Integration by Partial Fractions when the integrand is a rational function (a ratio of two polynomials) where the denominator can be factorised.
4. What is the correct approach for solving the Miscellaneous Exercise for Integrals in Chapter 7?
The Miscellaneous Exercise is designed to test your comprehensive understanding of the entire chapter. The best approach is to:
- Revise all concepts: Before attempting, thoroughly review all integration methods—substitution, by parts, partial fractions, and properties of definite integrals.
- Identify the core technique: Problems in this exercise often require a combination of techniques or a clever substitution. Analyse the problem carefully to identify the most efficient starting point.
- Don't give up: These questions are often more challenging and are excellent practice for higher-order thinking skill (HOTS) questions in board exams. Refer to step-by-step solutions to understand the logic if you get stuck.
5. Why is adding the constant of integration, '+ C', necessary in every indefinite integral?
The constant of integration, '+ C', is fundamental because integration is the reverse process of differentiation. The derivative of any constant (like 5, -10, or π) is zero. Therefore, when we find an antiderivative F(x) for a function f(x), we don't know if the original function had a constant term. The '+ C' represents this unknown constant and acknowledges that there is an entire family of functions (e.g., x² + 2, x² - 100) that have the same derivative.
6. What is the main difference between solving an indefinite integral and a definite integral?
The key difference lies in their results and interpretation. An indefinite integral, like ∫f(x)dx, gives a general function or a family of functions, F(x) + C, as its answer. In contrast, a definite integral, like ∫ₐᵇ f(x)dx, gives a single numerical value. This value is calculated using the Fundamental Theorem of Calculus by finding the antiderivative and evaluating it at the upper and lower limits (F(b) - F(a)).
7. I find trigonometric integrals confusing. What is a reliable strategy for integrals involving trigonometric identities?
The primary strategy for solving complex trigonometric integrals is to simplify the integrand before integrating. Your goal is to convert the expression into a form where standard integral formulas can be applied. Key tactics include:
- Using power-reducing formulas for terms like sin²(x) or cos²(x).
- Applying product-to-sum formulas for products like sin(ax)cos(bx).
- Making appropriate substitutions (e.g., u = sin(x) or u = tan(x)) after simplifying.
- Rewriting the integrand in terms of sin(x) and cos(x) if it involves other trig functions.
8. How do the NCERT Solutions for Class 12 Integrals help in preparing for board exams?
The NCERT Solutions for Class 12 Integrals are essential for board exam preparation because they demonstrate the exact step-by-step methodology that examiners expect. By following these solutions, you learn how to present your answers clearly, apply formulas correctly, and avoid common errors. Mastering them ensures you build a strong foundation that aligns perfectly with the CBSE 2025-26 syllabus, which is crucial for scoring full marks.
9. What is the learning objective of the 'method of inspection' in the first exercise of Integrals?
The 'method of inspection' serves a critical pedagogical purpose beyond just finding an answer. Its main objective is to build a deep, intuitive understanding of the inverse relationship between differentiation and integration. By asking you to guess the antiderivative, it forces you to think backwards: "Which function, when I differentiate it, gives me this result?" This strengthens your mental connection to derivative formulas and solidifies the core concept of an integral as an antiderivative.

















