CBSE Class 12 Maths Chapter 7: Integrals NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 12 PDF

This chapter provides essential NCERT Solutions for Class 12 Mathematics, focusing on the topic of Integrals (Chapter 7). It covers a wide array of fundamental integration formulas, including power rule, logarithmic, exponential, trigonometric, inverse trigonometric, and standard integral forms. The solutions also introduce techniques for integrating functions involving algebraic expressions like $ax+b$, quadratic expressions, and exponential functions multiplied by sine or cosine. Key formulas for integrals of square roots of quadratic expressions and rational functions are also detailed. These solutions are designed to help students understand the core concepts of integration, practice applying various formulas, and build a strong foundation for solving complex integration problems, which is crucial for exam preparation and higher mathematics.

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Quick info

Board	CBSE
Class	Class 12
Subject	गणित
Session	2026
Language	English
Type	NCERT Solutions
Chapter	7. समाकलन

Chapter summary

Chapter 7, Integrals, in the NCERT Solutions for Class 12 Maths, offers a comprehensive list of standard integral formulas. It covers basic power, exponential, logarithmic, and trigonometric integrals, along with those involving inverse trigonometric functions. The chapter also details integrals of the form $ax+b$, $\sqrt{a^2 \pm x^2}$, $\sqrt{x^2 - a^2}$, $\frac{1}{a^2 \pm x^2}$, and $\frac{1}{ax^2+bx+c}$. These solutions are vital for mastering the foundational techniques of integration.

Learning outcomes

Understand and apply the basic formulas for integration.
Identify and use standard integral forms for trigonometric and inverse trigonometric functions.
Apply integration formulas for linear expressions of the form (ax+b).
Utilize formulas for integrals involving square roots of quadratic expressions.
Solve integrals of rational functions using standard formulas.
Recognize and apply formulas for integrals of the form $\int e^{ax} \sin bx \, dx$ and $\int e^{ax} \cos bx \, dx$.

Topics covered

Paper topics

Basic Integration Formulas
Integration of Algebraic Functions
Integration of Trigonometric Functions
Integration of Exponential Functions
Integration of Logarithmic Functions
Integration of Inverse Trigonometric Functions
Integrals of the form (ax+b)^n
Integrals involving square roots of quadratic expressions
Integrals of rational functions
Integrals of e^(ax) sin(bx) and e^(ax) cos(bx)
Standard Integral Forms
Constant of Integration

Important topics

Basic Integration Formulas
Trigonometric Integrals
Integrals involving square roots of quadratic expressions
Integrals of rational functions
Standard Integral Forms
Constant of Integration

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Questions and Solutions

Question 1

Find the antiderivative (integral) of the function $\sin 2x$ using the method of inspection.

Solution:

To find the antiderivative of $\sin 2x$ by inspection, we need to think of a function whose derivative is $\sin 2x$. We know the derivative of $\cos 2x$ is $-2 \sin 2x$.

$\frac{d}{dx}(\cos 2x) = -2 \sin 2x$

Rearranging this, we get:

$\sin 2x = -\frac{1}{2} \frac{d}{dx}(\cos 2x)$

Using the property that if $f'(x) = g'(x)$, then $f(x) = g(x) + C$, we can integrate both sides with respect to x:

$\int \sin 2x \, dx = \int \left(-\frac{1}{2} \frac{d}{dx}(\cos 2x)\right) dx$

By the definition of an antiderivative, the integral of a derivative of a function is the function itself, plus a constant of integration.

$\int \sin 2x \, dx = -\frac{1}{2} \cos 2x + C$

Thus, the antiderivative of $\sin 2x$ is $-\frac{1}{2} \cos 2x + C$.

Common mistakes

Forgetting to add the constant of integration 'C'.
Errors in applying the power rule for integration, especially when n = -1.
Incorrectly recalling or applying formulas for trigonometric and inverse trigonometric integrals.
Mistakes in algebraic manipulation when simplifying expressions before integration.
Sign errors in the results of integration, particularly for trigonometric functions.

Revision tips

Memorize all standard integration formulas thoroughly.
Practice solving problems using each formula multiple times.
Pay close attention to the constant of integration 'C' in every indefinite integral.
Review the formulas for integrals involving square roots and rational functions, as they are frequently tested.
Work through the examples and exercises systematically to build confidence.

Practice MCQs

Q1. What is the integral of $x^n$ with respect to x, where $n $\neq$ -1$?

Q2. The integral of $ $\sin$ x$ with respect to x is:

Q3. Which formula represents the integral of $ $\frac{1}{\sqrt{1-x^2}}$ $ dx?

Q4. What is the integral of $e^x$ with respect to x?

Q5. The integral of $ $\sec$ ^2 x$ dx is:

Frequently asked questions

What is the main topic covered in CBSE Class 12 Maths Chapter 7?

Chapter 7, Integrals, covers the fundamental concepts and formulas of integration, which is the reverse process of differentiation.

Why is the constant of integration 'C' important in these solutions?

The constant of integration 'C' is added to every indefinite integral because the derivative of a constant is zero. It represents an arbitrary constant that accounts for all possible antiderivatives.

Are there specific formulas for integrating functions like $\sqrt{a^2 - x^2}$?

Yes, the NCERT Solutions provide standard formulas for integrals involving square roots of quadratic expressions, such as $\int \sqrt{a^2 - x^2} dx$, $\int \sqrt{a^2 + x^2} dx$, and $\int \sqrt{x^2 - a^2} dx$.

How do these NCERT Solutions help in exam preparation?

These solutions offer clear explanations and correct application of integration formulas, helping students build a strong foundation and practice effectively for their board exams.

What are the basic types of functions for which integration formulas are provided?

Formulas are provided for basic algebraic functions (like $x^n$), trigonometric functions (like $\sin x$, $\cos x$), exponential functions (like $e^x$), logarithmic functions (like $\frac{1}{x}$), and inverse trigonometric functions.

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.