NCERT Solutions for Class 12 Maths Chapter 7 Integrals: A Comprehensive Guide

The CBSE Class 12 Mathematics curriculum is crucial for students preparing for board exams and competitive exams like JEE, NEET, and other state-level entrance tests. Among the various chapters, Chapter 7 - Integrals is one of the most significant. It builds on the concepts learned in differentiation and covers integral calculus, which forms the foundation for higher mathematics, physics, engineering, and economics.

This article provides a detailed overview of the NCERT Solutions for Class 12 Maths Chapter 7 Integrals, ensuring that students have a strong grasp of each concept, topic, and exercise presented in the chapter.

Introduction to Integrals

Integral calculus is the branch of mathematics dealing with integrals, especially the methods of integration and their applications. It is essentially the reverse process of differentiation, often referred to as anti-differentiation. Chapter 7 in the NCERT Mathematics textbook for Class 12 thoroughly explores the concepts of integrals and provides a variety of solved examples and unsolved exercises for students to practice.

Key Topics Covered in Chapter 7 - Integrals

Chapter 7 is divided into several sections, each focusing on different types of integrals and their applications. Below is a breakdown of the key topics covered:

1. Introduction to Integrals
2. Integration as an Inverse Process of Differentiation
3. Methods of Integration
4. Integrals of Some Particular Functions
5. Integration by Partial Fractions
6. Integration by Parts
7. Definite Integrals
8. Fundamental Theorem of Calculus
9. Evaluation of Definite Integrals by Substitution
10. Some Properties of Definite Integrals

1. Introduction to Integrals

The chapter begins by introducing integrals as the inverse process of differentiation. If $f(x)$ is a function and its derivative is $f'(x)$, then the integral of $f'(x)$ gives back $f(x)$. This concept is the foundation of integral calculus and is crucial for understanding the remaining sections of the chapter.

2. Integration as an Inverse Process of Differentiation

Integration can be understood as finding a function whose derivative is given. This is referred to as the anti-derivative or primitive of the function. For instance, if the derivative of $F(x)$ is $f(x)$, then $F(x)$ is called an integral of $f(x)$. The process of finding integrals is known as integration.

The general form of an indefinite integral is given by:

$$\int f(x) \, dx = F(x) + C$$

where $C$ is the constant of integration.

3. Methods of Integration

Various methods are used for integrating different types of functions. The three main methods discussed in this chapter are:

• Integration by Substitution: This method is used when an integral can be simplified by a change of variables. It is useful when the integrand is a composite function.

• Integration by Partial Fractions: This method is applied to rational functions, where the numerator's degree is less than the denominator's degree. The integrand is expressed as a sum of simpler fractions, making it easier to integrate.

• Integration by Parts: This method is useful when the integrand is a product of two functions. It is based on the formula:

$$\int u \, dv = uv - \int v \, du$$

4. Integrals of Some Particular Functions

In this section, the chapter provides integrals for specific types of functions, such as trigonometric, exponential, and logarithmic functions. Memorizing these standard integrals is crucial for solving complex problems in competitive exams.

5. Integration by Partial Fractions

Integration by partial fractions is a technique used when dealing with rational functions, i.e., functions of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. The aim is to break down a complex fraction into simpler ones that can be integrated individually.

For example:

$$\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$$

Solving for $A$ and $B$ allows for easier integration.

6. Integration by Parts

Integration by parts is particularly useful when the integrand is a product of two functions. The formula is derived from the product rule of differentiation:

$$\int u \, dv = uv - \int v \, du$$

Choosing $u$ and $dv$ carefully is essential for simplifying the integral. The ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) helps in choosing $u$ and $dv$ correctly.

7. Definite Integrals

A definite integral has limits and represents the area under the curve from one point to another. It is given by:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$. Definite integrals are extensively used in physics, engineering, and probability.

8. Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, providing a way to evaluate definite integrals. It states:

1. If $f$ is continuous on $[a, b]$, then the function $g(x) = \int_{a}^{x} f(t) \, dt$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $g'(x) = f(x)$.

2. If $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

9. Evaluation of Definite Integrals by Substitution

This method involves changing the variable of integration to simplify the integral. For example, if $x = g(t)$, then:

$$\int f(x) \, dx = \int f(g(t)) \cdot g'(t) \, dt$$

This technique can significantly simplify complex integrals, especially when dealing with trigonometric and exponential functions.

10. Some Properties of Definite Integrals

The chapter also discusses several properties of definite integrals, such as:

• Additivity Property: If $a < c < b$, then:

$$\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$$

• Reversal Property:

$$\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$$

• Zero Property:

$$\int_{a}^{a} f(x) \, dx = 0$$

These properties are vital in solving complex problems and proving mathematical theorems.

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

The NCERT Solutions for Class 12 Maths Chapter 7 provide step-by-step explanations for each exercise in the chapter. Below is an overview of the exercises and their solutions:

Exercise 7.1: Introduction to Integration

This exercise focuses on the basics of integration as an inverse process of differentiation. Problems involve simple algebraic functions, and students are required to find the antiderivatives.

Exercise 7.2: Integration by Substitution

This exercise provides questions where substitution is necessary to simplify the integrand. It is crucial to identify the correct substitution to reduce the integral to a standard form.

Exercise 7.3: Integration Using Partial Fractions

Here, the integrands are rational functions that need to be expressed as partial fractions. Students learn to decompose the integrand and then integrate each term individually.

Exercise 7.4: Integration by Parts

Problems in this exercise involve integrating products of two functions. The challenge is to select $u$ and $dv$ appropriately to simplify the integration process.

Exercise 7.5: Definite Integrals

The focus here is on evaluating definite integrals using properties and the Fundamental Theorem of Calculus. Students also learn to apply substitution to definite integrals.

Exercise 7.6: Properties of Definite Integrals

This exercise emphasizes the use of various properties of definite integrals to simplify problems. It is essential to practice these properties to become adept at manipulating and solving integrals efficiently.

Why NCERT Solutions for Chapter 7 Are Important

• Foundation for Higher Studies: Understanding integrals is crucial for pursuing higher education in science, engineering, and mathematics.
• Competitive Exams: Integral calculus is a vital part of the syllabus for exams like JEE, NEET, and other state-level engineering and medical entrance exams.
• Step-by-Step Approach: The solutions provided in NCERT are detailed and help students understand the step-by-step process to solve complex problems.
• Concept Clarity: The solutions are designed to clarify concepts rather than just provide answers, ensuring a deeper understanding.

Conclusion

NCERT Solutions for Class 12 Maths Chapter 7 - Integrals are essential for mastering integral calculus. By following these solutions, students can gain a thorough understanding of integrals, which will not only help them excel in their board exams but also lay a strong foundation for future studies and competitive exams. It is recommended that students practice regularly, understand the core concepts, and utilize the properties of integrals to solve problems efficiently.