CBSE Class 12 Mathematics NCERT Solutions: Chapter 7 - Integrals

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NCERT Solutions PDF Class 12 PDF

This chapter provides essential NCERT Solutions for Class 12 Mathematics, focusing on the fundamental concepts of Integrals. It covers various techniques and applications of integration, which are crucial for understanding calculus. The solutions offer a detailed, step-by-step approach to solving problems related to finding antiderivatives of different functions. This resource is designed to help students grasp the core principles of integration, including the integration of trigonometric, exponential, and algebraic functions. By working through these solutions, students can build a strong foundation in calculus, which is vital for higher mathematics and related scientific fields. These solutions are an excellent tool for exam preparation, enabling students to revise and reinforce their understanding of integration methods and their applications, ensuring they are well-prepared for examinations.

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Quick info

BoardCBSE
ClassClass 12
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterIntegrals

Chapter summary

Chapter 7 on Integrals for Class 12 Mathematics introduces the concept of antiderivatives and the indefinite integral. The NCERT Solutions cover basic integration rules and the integration of standard functions like trigonometric, exponential, and polynomial functions. It lays the groundwork for more advanced integration techniques and applications, providing clear, step-by-step solutions to build student confidence and proficiency in calculus.

Learning outcomes

  • Understand the concept of an antiderivative and indefinite integral.
  • Apply basic integration rules to find antiderivatives of standard functions.
  • Solve problems involving the integration of trigonometric functions.
  • Solve problems involving the integration of exponential functions.
  • Solve problems involving the integration of polynomial functions.
  • Develop skills in finding the integral of combined functions.

Topics covered

Paper topics

  • Antiderivatives
  • Indefinite Integrals
  • Integration of Trigonometric Functions
  • Integration of Exponential Functions
  • Integration of Polynomial Functions
  • Linearity Property of Integrals
  • Basic Integration Formulas

Important topics

  • Understanding Antiderivatives
  • Integration of Standard Functions
  • Applying Linearity Property
  • Basic Integration Rules

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Questions and Solutions

Question 1

Find the antiderivative of the function: \sin 2x
Solution:

To find the antiderivative of \sin 2x, we need to find a function whose derivative is \sin 2x. We know the derivative of \cos 2x is \frac{d}{dx}(\cos 2x) = -2\sin 2x.

Rearranging this, we get \sin 2x = -\frac{1}{2} \frac{d}{dx} (\cos 2x).

By the property of integrals, we can bring the constant outside the derivative. Thus, \sin 2x = \frac{d}{dx} \left( -\frac{1}{2} \cos 2x \right).

Therefore, the antiderivative of \sin 2x is -\frac{1}{2}\cos 2x. We should also add the constant of integration, C, for indefinite integrals, making the general antiderivative -\frac{1}{2}\cos 2x + C.

Question 2

Find the antiderivative of the function: \cos 3x
Solution:

We are looking for a function whose derivative is \cos 3x. We know that the derivative of \sin 3x is \frac{d}{dx}(\sin 3x) = 3\cos 3x.

To isolate \cos 3x, we can rewrite the equation as \cos 3x = \frac{1}{3} \frac{d}{dx} (\sin 3x).

Using the constant multiple rule for differentiation, we can write this as \cos 3x = \frac{d}{dx} \left( \frac{1}{3} \sin 3x \right).

Hence, the antiderivative of \cos 3x is \frac{1}{3}\sin 3x. The general form including the constant of integration is \frac{1}{3}\sin 3x + C.

Question 3

Find the antiderivative of the function: e^{2x}
Solution:

We need to find a function whose derivative with respect to x is e^{2x}. Recall the derivative of an exponential function: \frac{d}{dx}(e^{2x}) = 2e^{2x}.

To find the antiderivative of e^{2x}, we rearrange the derivative formula: e^{2x} = \frac{1}{2} \frac{d}{dx} (e^{2x}).

Applying the constant multiple rule, we get e^{2x} = \frac{d}{dx} \left( \frac{1}{2} e^{2x} \right).

Therefore, the antiderivative of e^{2x} is \frac{1}{2}e^{2x}. The complete indefinite integral is \frac{1}{2}e^{2x} + C.

Question 4

Find the antiderivative of the function: (ax+b)^2
Solution:

We are looking for a function whose derivative is (ax+b)^2. Let's consider the derivative of (ax+b)^3. Using the chain rule, we have \frac{d}{dx}((ax+b)^3) = 3(ax+b)^2 \cdot \frac{d}{dx}(ax+b) = 3(ax+b)^2 \cdot a = 3a(ax+b)^2.

To find the antiderivative of (ax+b)^2, we rearrange this result: (ax+b)^2 = \frac{1}{3a} \frac{d}{dx} ((ax+b)^3).

Using the constant multiple rule, we can write this as (ax+b)^2 = \frac{d}{dx} \left( \frac{1}{3a} (ax+b)^3 \right).

Thus, the antiderivative of (ax+b)^2 is \frac{1}{3a}(ax+b)^3. The general indefinite integral is \frac{1}{3a}(ax+b)^3 + C.

Question 5

Find the antiderivative of the function: \sin 2x - 4e^{3x}
Solution:

To find the antiderivative of the expression \sin 2x - 4e^{3x}, we can integrate each term separately using the linearity property of integrals: \int (f(x) - g(x)) dx = \int f(x) dx - \int g(x) dx.

First, let's find the antiderivative of \sin 2x. As seen in Question 1, the antiderivative of \sin 2x is -\frac{1}{2}\cos 2x.

Next, let's find the antiderivative of 4e^{3x}. We know that the derivative of e^{3x} is 3e^{3x}. Therefore, the antiderivative of e^{3x} is \frac{1}{3}e^{3x}. Multiplying by 4, the antiderivative of 4e^{3x} is 4 \cdot \frac{1}{3}e^{3x} = \frac{4}{3}e^{3x}.

Combining these results, the antiderivative of \sin 2x - 4e^{3x} is -\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x}. Including the constant of integration, the general indefinite integral is -\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x} + C.

Common mistakes

  • Forgetting the constant of integration 'C'.
  • Errors in applying the power rule for integration.
  • Incorrectly differentiating when finding the antiderivative.
  • Mistakes in handling coefficients and constants within the integrand.

Revision tips

  • Review the basic differentiation rules before starting integration.
  • Practice identifying the correct integration formula for each function type.
  • Pay close attention to constants and coefficients during integration.
  • Always remember to add the constant of integration 'C' for indefinite integrals.
  • Work through each example and exercise step-by-step to ensure understanding.

Practice MCQs

Q1. What is the antiderivative of sin(2x)?

Q2. The integral of cos(3x) with respect to x is:

Q3. What is the antiderivative of e^(2x)?

Q4. The integral of (ax+b)^2 dx is:

Q5. What is the integral of sin(2x) - 4e^(3x) dx?

Frequently asked questions

What is an antiderivative in the context of Chapter 7 Integrals?

An antiderivative of a function f(x) is a function F(x) such that its derivative, F'(x), is equal to f(x). It's the reverse process of differentiation.

Why is the constant of integration 'C' important in indefinite integrals?

The constant of integration 'C' is added because the derivative of any constant is zero. Therefore, there are infinitely many antiderivatives for a given function, differing only by a constant.

How do these NCERT Solutions help with Class 12 Maths exam preparation?

These solutions provide clear, step-by-step explanations for each problem, helping students understand the methods and build confidence. They are essential for revising the concepts of integration effectively.

What types of functions are covered in the initial exercises of Chapter 7?

The initial exercises typically cover the integration of basic trigonometric functions (like sin(2x), cos(3x)), exponential functions (like e^(2x)), and polynomial functions (like (ax+b)^2).

Are the mathematical expressions in the questions and solutions preserved accurately?

Yes, all mathematical expressions, symbols, and equations from the original questions and solutions are preserved exactly as they appear in the source material.

Important links

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