NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 - Integrals

Edited By Ramraj Saini | Updated on Dec 03, 2023 09:41 PM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.1

NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 Integrals are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for Class 12 Maths chapter 7 exercise 7.1 is the first exercise of chapter 7 Integrals. Basic concepts of integrals are discussed in this exercise. This exercise includes concepts pertaining to finding out integrals of basic functions like sinx, cosx etc. Exercise 7.1 Class 12 Maths can be a good source to grasp the initial concepts of integrals. Head Start with basic concepts is a must in Integrals to reach an advanced level which can be easily learnt from NCERT Solutions for Class 12 Maths chapter 7 exercise 7.1 provided below. Also it is a good source to score well in CBSE Cass 12 Board Exam. Basic integration problems are also asked in competitive exams like JEE Main.

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In subsequent exercises of Class 11 Maths NCERT book, students will cover advanced problems related to finding the area of a curve etc. 12th class Maths exercise 7.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

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Access NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

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Integrals Class 12 Chapter 7 Exercise: 7.1

Question:1 Find an anti derivative (or integral) of the following functions by the method of inspection. $\sin 2x$

Answer:

GIven $\sin 2x$ ;

So, the anti derivative of $\sin 2x$ is a function of x whose derivative is $\sin 2x$ .

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

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

Therefore, we have $\implies \sin 2x = \frac{d}{dx}\left ( \frac{-1}{2}\cos 2x \right )$

Or, antiderivative of $\sin 2x$ is $\left ( \frac{-1}{2}\cos 2x \right )$ .

Question:2 Find an anti derivative (or integral) of the following functions by the method of inspection. $\cos 3x$

Answer:

GIven $\cos 3x$ ;

So, the antiderivative of $\cos 3x$ is a function of x whose derivative is $\cos 3x$ .

$\frac{d}{dx}\left ( \sin 3x \right ) = 3\cos3x$

$\implies \cos 3x =\frac{1}{3} \frac{d}{dx}\left ( \sin 3x \right )$

Therefore, we have the anti derivative of $\cos 3x$ is $\frac{1}{3}\sin 3x$ .

Question:3 Find an anti derivative (or integral) of the following functions by the method of inspection. $e ^{2x}$

Answer:

GIven $e ^{2x}$ ;

So, the anti derivative of $e ^{2x}$ is a function of x whose derivative is $e ^{2x}$ .

$\frac{d}{dx}\left ( e ^{2x}\right ) = 2e ^{2x}$

$\implies e ^{2x} = \frac{1}{2}\frac{d}{dx}(e ^{2x})$

$\therefore e ^{2x} = \frac{d}{dx}(\frac{1}{2}e ^{2x})$

Therefore, we have the anti derivative of $e^{2x}$ is $\frac{1}{2}e ^{2x}$ .

Question:4 Find an anti derivative (or integral) of the following functions by the method of inspection. $( ax + b )^2$

Answer:

GIven $( ax + b )^2$ ;

So, the anti derivative of $( ax + b )^2$ is a function of x whose derivative is $( ax + b )^2$ .

$\frac{d}{dx} (ax+b)^3 = 3a(ax+b)^2$

$\Rightarrow (ax+b)^2 =\frac{1}{3a}\frac{d}{dx}(ax+b)^3$

$\therefore (ax+b)^2 = \frac{d}{dx}[\frac{1}{3a}(ax+b)^3]$

Therefore, we have the anti derivative of $(ax+b)^2$ is $[\frac{1}{3a}(ax+b)^3]$ .

Question:5 Find an anti derivative (or integral) of the following functions by the method of inspection. $\sin 2x - 4 e ^{3x}$

Answer:

GIven $\sin 2x - 4 e ^{3x}$ ;

So, the anti derivative of

$\frac{d}{dx} (-\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x}) = \sin 2x -4e^{3x}$

Therefore, we have the anti derivative of $\sin 2x - 4 e ^{3x}$ is $\left ( -\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x} \right )$ .

Question:6 Find the following integrals

$\int ( 4e ^{3x}+1) dx$

Answer:

Given intergral $\int ( 4e ^{3x}+1) dx$ ;

$4\int e ^{3x} dx + \int 1 dx = 4\left ( \frac{e^{3x}}{3} \right ) +x +C$

or $\frac{4}{3} e^{3x} +x +C$ , where C is any constant value.

Question:7 Find the following integrals $\int x ^2 ( 1- \frac{1}{x^2})dx$

Answer:

Given intergral $\int x ^2 ( 1- \frac{1}{x^2})dx$ ;

$\int x^2 dx - \int1dx$

or $\frac{x^3}{3} - x +C$ , where C is any constant value.

Question:8 Find the following integrals $\int ( ax ^2 + bx + c ) dx$

Answer:

Given intergral $\int ( ax ^2 + bx + c ) dx$ ;

$\int ax^2\ dx + \int bx\ dx + \int c\ dx$

$= a\int x^2\ dx + b\int x\ dx + c\int dx$

$= a\frac{x^3}{3}+b\frac{x^2}{2}+cx +C$

or $\frac{ax^3}{3}+\frac{bx^2}{2}+cx +C$ , where C is any constant value.

Question:9 Find the following integrals intergration of $\int \left ( 2x^2 + e ^x \right ) dx$

Answer:

Given intergral $\int \left ( 2x^2 + e ^x \right ) dx$ ;

$\int 2x^2\ dx + \int e^{x}\ dx$

$= 2\int x^2\ dx + \int e^{x}\ dx$

$= 2\frac{x^3}{3}+e^{x} +C$

or $\frac{2x^3}{3}+e^{x} +C$ , where C is any constant value.

Question:10 Find the following integrals $\int \left ( \sqrt x - \frac{1}{\sqrt x } \right ) ^2 dx$

Answer:

Given integral $\int \left ( \sqrt x - \frac{1}{\sqrt x } \right ) ^2 dx$ ;

or $\int (x+\frac{1}{x}-2)\ dx$

$= \int x\ dx + \int \frac{1}{x}\ dx -2\int dx$

$= \frac{x^2}{2} + \ln|x| -2x +C$ , where C is any constant value.

Question:11 Find the following integrals intergration of $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$

Answer:

Given intergral $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$ ;

or $\int \frac{x^3}{x^2}\ dx+\int \frac{5x^2}{x^2}\ dx -4\int \frac{1}{x^2}\ dx$

$\int x\ dx + 5\int1. dx - 4\int x^{-2}\ dx$

$= \frac{x^2}{2}+5x-4\left ( \frac{x^{-1}}{-1} \right )+C$

Or, $\frac{x^2}{2}+5x+\frac{4}{x}+C$ , where C is any constant value.

Question:12 Find the following integrals $\int \frac{x^3+ 3x +4 }{\sqrt x } dx$

Answer:

Given intergral $\int \frac{x^3+ 3x +4 }{\sqrt x } dx$ ;

or $\int \frac{x^3}{x^{\frac{1}{2}}}\ dx+\int \frac{3x}{x^{\frac{1}{2}}}\ dx +4\int \frac{1}{x^{\frac{1}{2}}}\ dx$

$= \int x^{\frac{5}{2}}\ dx + 3\int x^{\frac{1}{2}}\ dx +4\int x^{-\frac{1}{2}}\ dx$

$=\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{3\left ( x^{\frac{3}{2}} \right )}{\frac{3}{2}}+\frac{4\left ( x^{\frac{1}{2}} \right )}{\frac{1}{2}} +C$

Or, $= \frac{2}{7}x^{\frac{7}{2}} +2x^{\frac{3}{2}}+8\sqrt{x} +C$ , where C is any constant value.

Question:13 Find the following integrals intergration of $\int \frac{x^3 - x^2 + x -1 }{x-1 } dx$

Answer:

Given integral $\int \frac{x^3 - x^2 + x -1 }{x-1 } dx$

It can be written as

$= \int \frac{x^2(x-1)+(x+1)}{(x-1)} dx$

Taking (x-1) common out

$= \int \frac{(x-1)(x^2+1)}{(x-1)} dx$

Now, cancelling out the term (x-1) from both numerator and denominator.

$= \int (x^2+1)dx$

Splitting the terms inside the brackets

$=\int x^2dx + \int 1dx$

$= \frac{x^3}{3}+x+c$

Question:14 Find the following integrals $\int (1-x) \sqrt x dx$

Answer:

Given intergral $\int (1-x) \sqrt x dx$ ;

$\int \sqrt{x}\ dx - \int x\sqrt{x}\ dx$ or

$\int x^{\frac{1}{2}}\ dx - \int x^{\frac{3}{2}} \ dx$

$= \frac{x^\frac{3}{2}}{\frac{3}{2}} - \frac{x^{\frac{5}{2}}}{\frac{5}{2}} +C$

$= \frac{2}{3}x^{\frac{3}{2}} - \frac{2}{5}x^{\frac{5}{2}}+C$ , where C is any constant value.

Question:15 Find the following integrals $\int \sqrt x ( 3x^2 + 2x +3 )dx$

Answer:

Given intergral $\int \sqrt x ( 3x^2 + 2x +3 )dx$ ;

$= \int 3x^2\sqrt{x}\ dx + \int 2x\sqrt{x}\ dx + \int 3\sqrt {x}\ dx$ or $= 3\int x^{\frac{5}{2}}\ dx + 2\int x^{\frac{3}{2}} \ dx +3\int x^{\frac{1}{2}} \ dx$

$= 3\frac{x^\frac{7}{2}}{\frac{7}{2}} +2\frac{x^{\frac{5}{2}}}{\frac{5}{2}} +3\frac{x^{\frac{3}{2}}}{\frac{3}{2}} +C$

$= \frac{6}{7}x^{\frac{7}{2}} + \frac{4}{5}x^{\frac{5}{2}}+ 2x^{\frac{3}{2}}+C$ , where C is any constant value.

Question:16 Find the following integrals $\int ( 2x - 3 \cos x + e ^x ) dx$

Answer:

Given integral $\int ( 2x - 3 \cos x + e ^x ) dx$ ;

splitting the integral as the sum of three integrals

$\int 2x\ dx -3 \int \cos x\ dx +\int e^{x}\ dx$

$= 2 \frac{x^2}{2} - 3 \sin x + e^x+C$

$= x^2 - 3 \sin x + e^x+C$ , where C is any constant value.

Question:17 Find the following integrals $\int ( 2 x ^2 - 3 \sin x + 5 \sqrt x ) dx$

Answer:

Given integral $\int ( 2 x ^2 - 3 \sin x + 5 \sqrt x ) dx$ ;

$2\int x^2\ dx -3\int \sin x\ dx + 5\int \sqrt {x}\ dx$

$= 2 \frac{x^3}{3} - 3(-\cos x ) +5\left ( \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ) +C$

$= \frac{2x^3}{3} +3\cos x +\frac{10}{3} x^{\frac{3}{2}}+C$ , where C is any constant value.

Question:18 Find the following integrals $\int \sec x ( \sec x + \tan x ) dx$

Answer:

Given integral $\int \sec x ( \sec x + \tan x ) dx$ ;

$\int (\sec^2x+ \sec x \tan x ) \ dx$

Using the integral of trigonometric functions

$= \int (sec^2 x )\ dx+ \int \sec x \tan x\ dx$

$= \tan x + \sec x +C$ , where C is any constant value.

Question:19 Find the following integrals intergration of $\int \frac{sec ^2 x }{cosec ^2 x } dx$

Answer:

Given integral $\int \frac{sec ^2 x }{cosec ^2 x } dx$ ;

$\int \frac{\frac{1}{\cos^2x}}{\frac{1}{\sin^2 x}}\ dx$

$= \int \frac{\sin^2 x }{\cos ^2 x } \ dx$

$=\int (\sec^2 x-1 )\ dx$

$=\int \sec^2 x\ dx-\int1 \ dx$

$= \tan x -x+C$ , where C is any constant value.

Question:20 Find the following integrals $\int \frac{2- 3 \sin x }{\cos ^ 2 x } dx$

Answer:

Given integral $\int \frac{2- 3 \sin x }{\cos ^ 2 x } dx$ ;

$\int \left ( \frac{2}{\cos^2x}-\frac{3\sin x }{\cos^2 x} \right )\ dx$

Using antiderivative of trigonometric functions

$= 2\tan x -3\sec x +C$ , where C is any constant value.

Question:21 Choose the correct answer
The anti derivative of $\left ( \sqrt x + 1/ \sqrt x \right )$ equals

$A) \frac{1}{3}x ^{1/3} + 2 x ^{1/2}+ C \\\\ B) \frac{2}{3}x ^{2/3} + \frac{1}{2}x ^{2}+ C \\\\ C ) \frac{2}{3}x ^{3/2} + 2 x ^{1/2}+ C\\\\ D) \frac{3}{2}x ^{3/2} + \frac{1}{2} x ^{1/2}+ C$

Answer:

Given to find the anti derivative or integral of $\left ( \sqrt x + 1/ \sqrt x \right )$ ;

$\int \left ( \sqrt x + 1/ \sqrt x \right )\ dx$

$\int x^{\frac{1}{2}}\ dx + \int x^{-\frac{1}{2}}\ dx$

$= \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+C$

$= \frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} +C$ , where C is any constant value.

Hence the correct option is (C).

Question:22 Choose the correct answer The anti derivative of

If $\frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4}$ such that f (2) = 0. Then f (x) is

$A ) x ^ 4 + \frac{1}{x^3} - \frac{129 }{8} \\\\ B ) x ^ 3 + \frac{1}{x^4} - \frac{129 }{8} \\\\ C ) x ^ 4 + \frac{1}{x^3} + \frac{129 }{8}\\\\ D) x ^ 3 + \frac{1}{x^4} - \frac{129 }{8}$

Answer:

Given that the anti derivative of $\frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4}$

So, $\frac{d}{dx}f(x) = 4 x ^3 - \frac{3}{x^4}$

$f(x) = \int 4 x ^3 - \frac{3}{x^4}\ dx$

$f(x) = 4\int x ^3 - 3\int {x^{-4}}\ dx$

$f(x) = 4\left ( \frac{x^4}{4} \right ) -3\left ( \frac{x^{-3}}{-3} \right )+C$

$f(x) = x^4+\frac{1}{x^3} +C$

Now, to find the constant C;

we will put the condition given, f (2) = 0

$f(2) = 2^4+\frac{1}{2^3} +C = 0$

$16+\frac{1}{8} +C = 0$

or $C = \frac{-129}{8}$

$\Rightarrow f(x) = x^4+\frac{1}{x^3}-\frac{129}{8}$

Therefore the correct answer is A .

More About NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

The NCERT Class 12 Maths chapter Integrals covers a total of 12 exercises including one miscellaneous exercise. Exercise 7.1 Class 12 Maths provides solutions to 22 main questions and their sub-questions. It includes basic questions related to finding integrals of basic functions. NCERT Solutions for Class 12 Maths chapter 7 exercise 7.1 must be referred to get a strong command on integrals.

Also Read| Integrals Class 12 Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1

The NCERT syllabus Class 12th Maths chapter 7 exercise provided here is in detail which is solved by subject matter experts .
Practicing exercise 7.1 Class 12 Maths can help students in a tremendous way to prepare for exams.
These Class 12 Maths chapter 7 exercise 7.1 solutions can be asked directly in the Board exams.
NCERT solutions for Class 12 Maths chapter 7 exercise 7.1 are highly recommended to students and can be used to solve physics questions also for the related concepts.

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Key Features Of NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7

Comprehensive Coverage: The solutions encompass all the topics covered in ex 7.1 class 12, ensuring a thorough understanding of the concepts.
Step-by-Step Solutions: In this class 12 maths ex 7.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
Accuracy and Clarity: Solutions for class 12 ex 7.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
Conceptual Clarity: In this 12th class maths exercise 7.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
Inclusive Approach: Solutions for ex 7.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
Relevance to Curriculum: The solutions for class 12 maths ex 7.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT Solutions Subject Wise

Subject Wise NCERT Exemplar Solutions

Happy learning!!!

Frequently Asked Questions (FAQs)

1. What is Integration ?

Integration is the process of finding the antiderivative of a function which gives the area of the curve formed by a function. The integration is the inverse process of differentiation.

2. Where Integration is used ?

Integration is used to find the area, volume etc. of many functions.

3. What is the importance of the Integrals chapter in Board exams ?

The Integral chapter covers a large portion of Class 12 Maths syllabus. It has applications in Physics as well. Hence it is advisable to students to not leave this chapter at any cost.

4. What is the difficulty level of questions of this chapter?

In board, easy to moderate level of questions are asked but in competitive exams, advanced level of problems are asked which can only be tackled with practice.

5. Mention some topics in Exercise 7.1 Class 12 Maths.

It includes basic topics related to finding the integration of basic functions. Eg sinx, cosx etc.

6. Mention the total number of questions in this exercise ?

There are 22 questions in this exercise. For more questions students can refer to NCERT exemplar problems

Also Read

NCERT Class 12 Maths Notes - Download Chapter wise Free PDFs

NCERT Class 12 Physics Chapter 9 Notes Ray Optics and Optical Instruments - Download PDF

NCERT Class 12 Physics Chapter 8 Notes Electromagnetic Waves - Download PDF

NCERT Class 12 Physics Chapter 7 Notes Alternating Current - Download PDF

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NCERT Books for Class 12 - Download All Subjects PDFs Here

NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves

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Questions related to CBSE Class 12th

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quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

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Jefferson 25 September,2024

i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

Re-evaluate Your Study Strategies:
- Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
- Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
- Practice Regularly: Consistent practice is key to mastering chemistry.
Consider Professional Help:
- Tutoring: A tutor can provide personalized guidance and support.
- Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.
Explore Alternative Options:
- Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
- Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.
Focus on NEET 2025 Preparation:
- Stay Dedicated: Continue your NEET preparation with renewed determination.
- Utilize Resources: Make use of study materials, online courses, and mock tests.
Seek Support:
- Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
- Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.

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Kanishka kaushikii 17 September,2024

i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

Type A: For candidates scoring 60% or above in the exam.

Type B: For candidates scoring between 50% and 60%.

Type C: For candidates scoring between 40% and 50%.

Cash Scholarship:

Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.

Hope you find this useful!

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Yuvan 30 August,2024

i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024

Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

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kumardurgesh1802 10 July,2024

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Central Board of Secondary Education Class 12th Examination

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

NCERT Solutions for Exercise 7.1 Class 12 Maths Chapter 7 - Integrals

NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.1

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Answer:

Question:2 Find an anti derivative (or integral) of the following functions by the method of inspection.

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Question:3 Find an anti derivative (or integral) of the following functions by the method of inspection.

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Question:4 Find an anti derivative (or integral) of the following functions by the method of inspection.

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Question:5 Find an anti derivative (or integral) of the following functions by the method of inspection.

Answer:

Question:6 Find the following integrals

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Question:7 Find the following integrals

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Question:8 Find the following integrals

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Question:9 Find the following integrals intergration of

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Question:10 Find the following integrals

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Question:11 Find the following integrals intergration of

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Question:12 Find the following integrals

Answer:

Question:13 Find the following integrals intergration of

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Question:14 Find the following integrals

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Question:15 Find the following integrals

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Question:16 Find the following integrals

Answer:

Question:17 Find the following integrals

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Question:18 Find the following integrals

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Question:19 Find the following integrals intergration of

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Question:20 Find the following integrals

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Question:21 Choose the correct answer The anti derivative of equals

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Question:22 Choose the correct answer The anti derivative of

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Question:2 Find an anti derivative (or integral) of the following functions by the method of inspection. $\cos 3x$

Question:3 Find an anti derivative (or integral) of the following functions by the method of inspection. $e ^{2x}$

Question:4 Find an anti derivative (or integral) of the following functions by the method of inspection. $( ax + b )^2$

Question:5 Find an anti derivative (or integral) of the following functions by the method of inspection. $\sin 2x - 4 e ^{3x}$

Question:7 Find the following integrals $\int x ^2 ( 1- \frac{1}{x^2})dx$

Question:8 Find the following integrals $\int ( ax ^2 + bx + c ) dx$

Question:9 Find the following integrals intergration of $\int \left ( 2x^2 + e ^x \right ) dx$

Question:10 Find the following integrals $\int \left ( \sqrt x - \frac{1}{\sqrt x } \right ) ^2 dx$

Question:11 Find the following integrals intergration of $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$

Question:12 Find the following integrals $\int \frac{x^3+ 3x +4 }{\sqrt x } dx$

Question:13 Find the following integrals intergration of $\int \frac{x^3 - x^2 + x -1 }{x-1 } dx$

Question:14 Find the following integrals $\int (1-x) \sqrt x dx$

Question:15 Find the following integrals $\int \sqrt x ( 3x^2 + 2x +3 )dx$

Question:16 Find the following integrals $\int ( 2x - 3 \cos x + e ^x ) dx$

Question:17 Find the following integrals $\int ( 2 x ^2 - 3 \sin x + 5 \sqrt x ) dx$

Question:18 Find the following integrals $\int \sec x ( \sec x + \tan x ) dx$

Question:19 Find the following integrals intergration of $\int \frac{sec ^2 x }{cosec ^2 x } dx$

Question:20 Find the following integrals $\int \frac{2- 3 \sin x }{\cos ^ 2 x } dx$

Question:21 Choose the correct answer
The anti derivative of $\left ( \sqrt x + 1/ \sqrt x \right )$ equals