Open App

JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > NCERT Solutions Exercise- 7.3: Integrals

NCERT Solutions Class 12 Maths Chapter 7 - Integrals

Download, print and study this document offline

Please wait while the PDF view is loading

 Page 1


1 	 / 	 2 1
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 3
F i n d 	 t h e 	 i n t e g r a l s 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	
A n s . 	
= 	
U s i n g 	
= 	 	
= 	 	 	 	 	 	
U s i n g
= 	 	 A n s .
2 . 	
A n s . 	 	 	 = 	 	
= 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n B 	 c o s A = s i n ( A + B ) - s i n ( A - B )
Page 2


1 	 / 	 2 1
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 3
F i n d 	 t h e 	 i n t e g r a l s 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	
A n s . 	
= 	
U s i n g 	
= 	 	
= 	 	 	 	 	 	
U s i n g
= 	 	 A n s .
2 . 	
A n s . 	 	 	 = 	 	
= 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n B 	 c o s A = s i n ( A + B ) - s i n ( A - B )
2 	 / 	 2 1
= 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
= 	 	
= 	 	
= 	 	 A n s .
3 . 	 	
A n s . 	 	
= 	
= 	 	 	 	 	 	
	 U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
= 	 	
= 	 	 	 	 	 	 	 	
= 	 	 	 	
U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
a n d 	 	 	 	 	 	 	 	
= 	 	
Page 3


1 	 / 	 2 1
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 3
F i n d 	 t h e 	 i n t e g r a l s 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	
A n s . 	
= 	
U s i n g 	
= 	 	
= 	 	 	 	 	 	
U s i n g
= 	 	 A n s .
2 . 	
A n s . 	 	 	 = 	 	
= 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n B 	 c o s A = s i n ( A + B ) - s i n ( A - B )
2 	 / 	 2 1
= 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
= 	 	
= 	 	
= 	 	 A n s .
3 . 	 	
A n s . 	 	
= 	
= 	 	 	 	 	 	
	 U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
= 	 	
= 	 	 	 	 	 	 	 	
= 	 	 	 	
U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
a n d 	 	 	 	 	 	 	 	
= 	 	
3 	 / 	 2 1
= 	 	 	 	 	 	 	 A n s .
4 . 	
A n s . 	 	
= 	 	
	
= 	 	
= 	 	
= 	 	
= 	
A n o t h e r 	 M e t h o d 	 	 	
	 	 	 	 	
U s i n g 	
	 	 	 	 l e t 	 	
	 	 t h e r e f o r e 	 	
Page 4


1 	 / 	 2 1
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 3
F i n d 	 t h e 	 i n t e g r a l s 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	
A n s . 	
= 	
U s i n g 	
= 	 	
= 	 	 	 	 	 	
U s i n g
= 	 	 A n s .
2 . 	
A n s . 	 	 	 = 	 	
= 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n B 	 c o s A = s i n ( A + B ) - s i n ( A - B )
2 	 / 	 2 1
= 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
= 	 	
= 	 	
= 	 	 A n s .
3 . 	 	
A n s . 	 	
= 	
= 	 	 	 	 	 	
	 U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
= 	 	
= 	 	 	 	 	 	 	 	
= 	 	 	 	
U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
a n d 	 	 	 	 	 	 	 	
= 	 	
3 	 / 	 2 1
= 	 	 	 	 	 	 	 A n s .
4 . 	
A n s . 	 	
= 	 	
	
= 	 	
= 	 	
= 	 	
= 	
A n o t h e r 	 M e t h o d 	 	 	
	 	 	 	 	
U s i n g 	
	 	 	 	 l e t 	 	
	 	 t h e r e f o r e 	 	
4 	 / 	 2 1
	 	 	 	 t h u s 	
	 	 	 	 A n s
5 . 	
A n s . 	 	
= 	 	
= 	
= 	
= 	 	
= 	 	
= 	 	
= 	 	
= 	
Page 5


1 	 / 	 2 1
N C E R T 	 s o l u t i o n
C h a p t e r 	 - 	 7
I n t e g r a l s 	 - 	 E x e r c i s e 	 7 . 3
F i n d 	 t h e 	 i n t e g r a l s 	 o f 	 t h e 	 f o l l o w i n g 	 f u n c t i o n s 	 i n 	 E x e r c i s e s 	 1 	 t o 	 9 .
1 . 	
A n s . 	
= 	
U s i n g 	
= 	 	
= 	 	 	 	 	 	
U s i n g
= 	 	 A n s .
2 . 	
A n s . 	 	 	 = 	 	
= 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n B 	 c o s A = s i n ( A + B ) - s i n ( A - B )
2 	 / 	 2 1
= 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
= 	 	
= 	 	
= 	 	 A n s .
3 . 	 	
A n s . 	 	
= 	
= 	 	 	 	 	 	
	 U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
= 	 	
= 	 	 	 	 	 	 	 	
= 	 	 	 	
U s i n g 	 2 c o s A 	 c o s B = c o s ( A + B ) + c o s ( A - B )
a n d 	 	 	 	 	 	 	 	
= 	 	
3 	 / 	 2 1
= 	 	 	 	 	 	 	 A n s .
4 . 	
A n s . 	 	
= 	 	
	
= 	 	
= 	 	
= 	 	
= 	
A n o t h e r 	 M e t h o d 	 	 	
	 	 	 	 	
U s i n g 	
	 	 	 	 l e t 	 	
	 	 t h e r e f o r e 	 	
4 	 / 	 2 1
	 	 	 	 t h u s 	
	 	 	 	 A n s
5 . 	
A n s . 	 	
= 	 	
= 	
= 	
= 	 	
= 	 	
= 	 	
= 	 	
= 	
5 	 / 	 2 1
A n o t h e r 	 m e t h o d 	 	 	 l e t 	 	 	 I = 	 	 	 	
l e t 	 c o s x = t
- s i n x 	 d x = d t
s i n x 	 	 d x 	 = 	 - d t
	 	 a n s .
	
6 . 	 	
A n s . 	 	
= 	 	
= 	 	 	 	 	 	 	 	 	 	 U s i n g 	 2 s i n 	 A 	 s i n 	 B = c o s 	 ( A - B ) 	 - 	 c o s 	 ( 	 A + B )
	 	
= 	 	 	
u s i n g 	 2 c o s A 	 s i n B = s i n ( A + B ) - s i n ( A - B )

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|467 docs\|187 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|467 docs|187 tests

Join Course for Free

FAQs on NCERT Solutions Class 12 Maths Chapter 7 - Integrals

1. What are integrals?

Ans. Integrals are mathematical tools used to calculate the area under a curve or to find the accumulation of quantities over a given interval. They are an important concept in calculus and have applications in various fields such as physics, engineering, and economics.

2. What is the significance of integrals?

Ans. Integrals have several significant applications. They can be used to find the area between curves, calculate volumes of irregular shapes, determine the average value of a function, solve differential equations, and analyze rates of change. They are fundamental in understanding the concept of calculus and its real-world applications.

3. How do I solve integrals?

Ans. To solve integrals, you need to determine the antiderivative of the function to be integrated. This involves finding a function whose derivative is equal to the given function. Once you have the antiderivative, you can evaluate the integral by plugging in the limits of integration and subtracting the value at the lower limit from the value at the upper limit.

4. What are definite integrals?

Ans. Definite integrals are a type of integral that have specific limits of integration. They represent the accumulation of quantities over a given interval. The value of a definite integral gives the net signed area between the curve and the x-axis within the specified limits.

5. Are there any techniques to simplify integral calculations?

Ans. Yes, there are several techniques to simplify integral calculations. Some common methods include integration by substitution, integration by parts, partial fraction decomposition, and trigonometric substitutions. These techniques help to transform complex integrals into simpler forms, making them easier to solve.

Related Exams

JEE Commerce Grade 12 Humanities/Arts Airforce X Y / Indian Navy SSR GRE Teaching

About this Document

2.9K Views

4.78/5 Rating

Feb 16, 2025 Last updated

Document Description: NCERT Solutions Exercise- 7.3: Integrals for JEE 2025 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for NCERT Solutions Exercise- 7.3: Integrals have been prepared according to the JEE exam syllabus. Information about NCERT Solutions Exercise- 7.3: Integrals covers topics like and NCERT Solutions Exercise- 7.3: Integrals Example, for JEE 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for NCERT Solutions Exercise- 7.3: Integrals.

Introduction of NCERT Solutions Exercise- 7.3: Integrals in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & NCERT Solutions Exercise- 7.3: Integrals in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: NCERT Solutions Class 12 Maths Chapter 7 - Integrals

Description

Full syllabus notes, lecture & questions for NCERT Solutions Class 12 Maths Chapter 7 - Integrals - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download

Information about NCERT Solutions Exercise- 7.3: Integrals

In this doc you can find the meaning of NCERT Solutions Exercise- 7.3: Integrals defined & explained in the simplest way possible. Besides explaining types of NCERT Solutions Exercise- 7.3: Integrals theory, EduRev gives you an ample number of questions to practice NCERT Solutions Exercise- 7.3: Integrals tests, examples and also practice JEE tests

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|467 docs\|187 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|467 docs|187 tests

Join Course for Free

Download as PDF

Explore Courses for JEE exam

NCERT Solutions Class 12 Maths Chapter 7 - Integrals

Extra Questions

study material

Free

NCERT Solutions Class 12 Maths Chapter 7 - Integrals

shortcuts and tricks

MCQs

Summary

video lectures

practice quizzes

Sample Paper

Semester Notes

Exam

Objective type Questions

mock tests for examination

Viva Questions

past year papers

Previous Year Questions with Solutions

ppt

pdf

Important questions

;

Additional Information about NCERT Solutions Exercise- 7.3: Integrals for JEE Preparation

NCERT Solutions Exercise- 7.3: Integrals Free PDF Download

The NCERT Solutions Exercise- 7.3: Integrals is an invaluable resource that delves deep into the core of the JEE exam. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. With the help of these notes, you can grasp complex subjects quickly, revise important points easily, and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner, allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material, or toppers' notes, this PDF has got you covered. Download the NCERT Solutions Exercise- 7.3: Integrals now and kickstart your journey towards success in the JEE exam.

Importance of NCERT Solutions Exercise- 7.3: Integrals

The importance of NCERT Solutions Exercise- 7.3: Integrals cannot be overstated, especially for JEE aspirants. This document holds the key to success in the JEE exam. It offers a detailed understanding of the concept, providing invaluable insights into the topic. By knowing the concepts well in advance, students can plan their preparation effectively. Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.

NCERT Solutions Exercise- 7.3: Integrals Notes

NCERT Solutions Exercise- 7.3: Integrals Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to NCERT Solutions Exercise- 7.3: Integrals. It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation. Practice papers and question papers enable you to assess your progress effectively. Additionally, the paper analysis provides valuable tips for tackling the exam strategically. Access to Toppers' notes gives you an edge in understanding complex concepts. Whether you're a beginner or aiming for advanced proficiency, NCERT Solutions Exercise- 7.3: Integrals Notes on EduRev are your ultimate resource for success.

NCERT Solutions Exercise- 7.3: Integrals JEE Questions

The "NCERT Solutions Exercise- 7.3: Integrals JEE Questions" guide is a valuable resource for all aspiring students preparing for the JEE exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

Study NCERT Solutions Exercise- 7.3: Integrals on the App

Students of JEE can study NCERT Solutions Exercise- 7.3: Integrals alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the NCERT Solutions Exercise- 7.3: Integrals, students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc. The EduRev App will make your learning easier as you can access it from anywhere you want. The content of NCERT Solutions Exercise- 7.3: Integrals is prepared as per the latest JEE syllabus.

Education Revolution