NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals
NCERT solutions for class 12 maths chapter 7 miscellaneous exercise provides questions based on integration of square root functions, trigonometric functions etc. Class 12 maths chapter 7 miscellaneous exercise is last but not the least as you can find some of the questions in previous years from this exercise only. class 12 maths chapter 7 miscellaneous exercise can be found difficult for some students but after giving sufficient time, you will be able to deal with the questions. NCERT solutions for class 12 maths chapter 7 with all the other exercises can be found in NCERT Class 12th book. Apart from class 12 maths chapter 7 miscellaneous exercise solutions, you can also refer below exercises for wide understanding of the topic.
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.1
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.2
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.3
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.4
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.5
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.6
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.7
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.8
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.9
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.10
NCERT solutions for class 12 maths chapter 7 Integrals Exercise 7.11
NCERT solutions for class 12 maths chapter 7 Integrals-Miscellaneous Exercise
Question:1 Integrate the functions in Exercises 1 to 24.
Answer:
Firstly we will simplify the given equation :-
Let
By solving the equation and equating the coefficients of x 2 , x and the constant term, we get
Thus the integral can be written as :
or
Question:2 Integrate the functions in Exercises 1 to 24.
Answer:
At first we will simplify the given expression,
or
Now taking its integral we get,
or
or
Question:3 Integrate the functions in Exercises 1 to 24.
Answer:
Let
Using the above substitution we can write the integral is
or
or
or
or
Question:4 Integrate the functions in Exercises 1 to 24.
Answer:
For the simplifying the expression, we will multiply and dividing it by x -3 .
We then have,
Now, let
Thus,
or
Question:5 Integrate the functions in Exercises 1 to 24.
Answer:
Put
We get,
or
or
or
Now put in the above result :
Question:6 Integrate the functions in Exercises 1 to 24.
Answer:
Let us assume that :
Solving the equation and comparing coefficients of x 2 , x and the constant term.
We get,
Thus the equation becomes :
or
or
or
or
Question:7 Integrate the functions in Exercises 1 to 24.
Answer:
We have,
Assume :-
Putting this in above integral :
or
or
or
or
Question: Integrate the functions in Exercises 1 to 24.
Answer:
We have the given integral
Assume
So, this substitution gives,
or
Question:10 Integrate the functions in Exercises 1 to 24.
Answer:
We have
Simplifying the given expression, we get :
or
or
or
Thus,
and
Question:11 Integrate the functions in Exercises 1 to 24.
Answer:
For simplifying the given equation, we need to multiply and divide the expression by .
Thus we obtain :
or
or
or
Thus integral becomes :
or
or
Question:12 Integrate the functions in Exercises 1 to 24.
Answer:
Given that to integrate
Let
the required solution is
Question:13 Integrate the functions in Exercises 1 to 24.
Answer:
we have to integrate the following function
Let
using this we can write the integral as
Question:14 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
Let
Now, Using partial differentiation,
Equating the coefficients of and constant value,
A + C = 0 C = -A
B + D = 0 B = -D
4A + C =0 4A = -C
4A = A
A = 0 = C
4B + D = 1 4B – B = 1
B = 1/3 = -D
Putting these values in equation, we have
Question:15 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
(let)
Let
using the above substitution the integral is written as
Question:16 Integrate the functions in Exercises 1 to 24.
Answer:
Given the function to be integrated as
Let
Let
Question:17 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
Let
Let f(ax +b) = t ⇒ a .f ' (ax + b)dx = dt
Now we can write the ntegral as
Question:18 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
Let
We know the identity that
sin (A+B) = sin A cos B + cos A sin B
Question:19 Integrate the functions in Exercises 1 to 24.
Answer:
We have
or
or
or
or
or
Thus
Now we will solve I'.
Put x = t 2 .
Differentiating the equation wrt x, we get
Thus
or
Using integration by parts, we get :
or
We know that
Thus it becomes :
So I come to be :-
Question:20 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
= I (let)
Let
using the above substitution we can write the integral as
Question:21 Integrate the functions in Exercises 1 to 24.
Answer:
Given to evaluate
now the integral becomes
Let tan x = f(x)
Question:22 Integrate the functions in Exercises 1 to 24.
Answer:
Given,
using partial fraction we can simplify the integral as
Let
Equating the coefficients of x, x 2 and constant value, we get:
A + C = 1
3A + B + 2C = 1
2A+2B+C =1
Solving these:
A= -2, B=1 and C=3
Question:23 Integrate the functions in Exercises 1 to 24.
Answer:
We have
Let us assume that :
Differentiating wrt x,
Substituting this in the original equation, we get
or
or
Using integration by parts , we get
or
or
Putting all the assumed values back in the expression,
or
Question:24 Integrate the functions in Exercises 1 to 24.
Answer:
Here let's first reduce the log function.
Now, let
So our function in terms if new variable t is :
now let's solve this By using integration by parts
Question:25 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Since, we have multiplied by some function, let's try to make that function in any function and its derivative.Basically we want to use the property,
So,
Here let's use the property
so,
Question:26 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
First, let's convert sin and cos into tan and sec. (because we have a good relation in tan and square of sec,)
Let' divide both numerator and denominator by
Now lets change the variable
the limits will also change since the variable is changing
So, the integration becomes:
Question:27 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Lets first simplify the function.
As we have a good relation in between squares of the tan and square of sec lets try to take our equation there,
AS we can write square of sec in term of tan,
Now let's calculate the integral of the second function, (we already have calculated the first function)
let
here we are changing the variable so we have to calculate the limits of the new variable
when x = 0, t = 2tanx = 2tan(0)=0
when
our function in terms of t is
Hence our total solution of the function is
Question:28 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Here first let convert sin2x as the angle of x ( sinx, and cosx)
Now let's remove the square root form function by making a perfect square inside the square root
Now let
,
since we are changing the variable, limit of integration will change
our function in terms of t :
Question:29 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
First, let's get rid of the square roots from the denominator,
Question:30 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
First let's assume t = cosx - sin x so that (sinx +cosx)dx=dt
So,
Now since we are changing the variable, the new limit of the integration will be,
when x = 0, t = cos0-sin0=1-0=1
when
Now,
Hence our function in terms of t becomes,
Question:31 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Let I =
Here, we can see that if we put sinx = t, then the whole function will convert in term of t with dx being changed to dt.so
Now the important step here is to change the limit of the integration as we are changing the variable.so,
So our function becomes,
Now, let's integrate this by using integration by parts method,
Question:32 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Let I = -(i)
Replacing x with ( -x),
- (ii)
Adding (i) and (ii)
Question:33 Evaluate the definite integrals in Exercises 25 to 33.
Answer:
Given integral
So, we split it in according to intervals they are positive or negative.
Now,
as
is positive in the given x -range
Therefore,
as
is in the given x -range
and
in the range
Therefore,
as
is in the given x -range
and
in the range
Therefore,
So, We have the sum
Question:34 Prove the following (Exercises 34 to 39)
.
Answer:
L.H.S =
We can write the numerator as [(x+1) -x]
= RHS
Hence proved.
Question:35 Prove the following (Exercises 34 to 39)
Answer:
Integrating I by parts
Applying Limits from 0 to 1
Hence proved I = 1
Question:36 Prove the following (Exercises 34 to 39)
Answer:
The Integrand g(x) therefore is an odd function and therefore
Question:37 Prove the following (Exercises 34 to 39)
Answer:
For I 2 let cosx=t, -sinxdx=dt
The limits change to 0 and 1
I 1 -I 2 =2/3
Hence proved.
Question:38 Prove the following (Exercises 34 to 39)
Answer:
The integral is written as
Hence Proved
Question:39 Prove the following (Exercises 34 to 39)\
Answer:
Integrating by parts we get
For I 2 take 1-x 2 = t 2 , -xdx=tdt
Hence Proved
Question:40 Evaluate
as a limit of a sum.
Answer:
As we know
where b-a=hn
In the given problem b=1, a=0 and
Question:41 Choose the correct answers in Exercises 41 to 44.
Answer:
the above integral can be re arranged as
let e x =t, e x dx=dt
(A) is correct
Question:42 Choose the correct answers in Exercises 41 to 44.
Answer:
cos2x=cos 2 x-sin 2 x
let sinx+cosx=t,(cosx-sinx)dx=dt
hence the given integral can be written as
B is correct
Question:43 Choose the correct answers in Exercises 41 to 44.
Answer:
As we know
Using the above property we can write the integral as
Answer (D) is correct
Question: 44 Choose the correct answers in Exercises 41 to 44.
Answer:
as
Now the integral can be written as
(B) is correct.
More about NCERT solutions for class 12 maths chapter 7 miscellaneous exercise
The NCERT class 12 maths chapter integrals is most important in the whole maths syllabus and it has applications in Physics and chemistry also. NCERT solutions for class 12 maths chapter 7 miscellaneous exercise deals with quite tricky and interesting questions which can be realised while solving the questions. NCERT solutions for class 12 maths chapter 7 miscellaneous exercise will take some longer time to complete but one should remain focussed while solving it as it is quite important. class 12 maths chapter 7 miscellaneous solutions is a good source to practice well.
Benefits of ncert solutions for class 12 maths chapter 7 miscellaneous exercises
The class 12th maths chapter 7 exercise has great importance and it should be done in step by step manner.
Class 12 maths chapter 7 miscellaneous exercises Will take more effort than other exercises, Hence should be done patiently.
These class 12 maths chapter 7 miscellaneous exercises along with previous exercises are good to go for the exam.
Also see-
NCERT Solutions Subject Wise
Subject wise NCERT Exemplar solutions
Happy learning!!!
Frequently Asked Question (FAQs) - NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals
Question: Can this exercise be done with self study only ?
Answer:
Yes, but for that basic concepts must be clear.
Question: What is the weightage from miscellaneous exercise in the examination ?
Answer:
Around 5 marks of questions are asked in the examination.
Question: Is it mandatory to solve every question of Miscellaneous exercise ?
Answer:
No, but questions on different concepts must be solved.
Question: Can shortcuts be used while solving the questions.?
Answer:
In boards, step by step method is used but shortcuts can be used in JEE and NEET.
Question: How much time will it take topcomlete miscellaneous exercise for the first time ?
Answer:
It can take 3 to 4 hours for the first time.
Question: What is the level of questions in Miscellaneous exercise ?
Answer:
Advanced level problems are dealt with in this exercise.
Latest Articles
National Scholarship Portal 2022 - Check NSP 2.0 Last Date, On...
National Scholarship Portal 2022 - NSP is the central portal t...
TOSS Result 2023 for SSC & Inter - Telangana Open School Socie...
TOSS Result 2023 - Telangana Open School Society releases the ...
Maharashtra HSC Result 2023 - Check Maharashtra board 12th res...
Maharashtra HSC Result 2023 - Maharashtra Board will announce ...
CBSE 12th Result 2023 - CBSE Class 12th Result at cbseresults....
CBSE Class 12 Result 2023 - Central Board of Secondary Educati...
IEO Result 2022-23 (Released)- Check SOF IEO result at sofworl...
IEO Result 2022-23 - Know where to check SOF IEO result 2022-2...
NSO Result 2022-23 - Check SOF NSO Result for Level 1 & 2 Here
NSO Result 2022-23 - Science Olympiad Foundation has released ...
IMO Results 2022-23 (Released) - Check SOF IMO Level 1 & 2 Res...
IMO Results 2022-23 - Science Olympiad Foundation has released...
TN 12th Result 2023 - Check Tamil Nadu Class 12 Result for Sci...
TN 12th Result 2023 - Directorate of Government Examinations r...
TN 11th Result 2023 - Plus One Tamil Nadu Result @tnresults.ni...
TN 11th Result 2023 Tamilnadu - DGE will release the Tamil Nad...
CGBSE 12th Result 2023 - Chhattisgarh Board Class 12th Result ...
CGBSE 12th Result 2023 - Chhattisgarh Board will release the C...