NCERT Solutions for Class 12 Maths Chapter 7 Integrals
NCERT solutions for class 12 maths chapter 7 Integrals: The word integration literally means summation. When you have to find the sum of finite numbers you can do by simply adding these numbers. But when you are finding the sum of a certain number of elements as the number of elements tends to infinity and at the same time each term becomes infinitesimally small, you can use a prosses to find its limit called integration. In this article, you will find NCERT solutions for class 12 maths chapter 7 integrals which is very helpful when you are solving NCERT textbook questions. In the differential calculus you learnt about the differentiation, defining tangent and and how to calculate the slope of the line. Integration is the inverse process of differentiation. Remember every function is not integrable which means you can't integrate every function. The function is integrable only if the function is already differentiated. In the NCERT solutions for class 12 maths chapter 7 integrals, you will learn the different methods of integration for different types of functions. NCERT solutions for class 12 maths chapter 7 integrals are very important from the 12th CBSE board exam point of view and are also important for competitive examinations like JEE Main, VITEEE, BITSAT, etc. Here you will find all NCERT solutions from class 6 to 12 at a single place to help you to learn CBSE maths and science. Here you will get NCERT solutions for class 12 also.
Integrals has 13 % weightage in 12 board final examination. Next chapter "applications of integrals" is also dependent on this chapter. So you should try to solve every problem of this chapter on your own. If you are not able to do, you can take the help of these NCERT solutions for class 12 maths chapter 7 integrals. In this chapter, there are 11 exercises with 227 questions and also 44 questions are there in miscellaneous exercise. Here, the NCERT solutions for class 12 maths chapter 7 integrals are solved and explained in detail to develop a grip on the topic. Here, you will learn two types of integrals: Definite integral and Indefinite integral and also learn their properties and formulas.
Definite Integral  Indefinite Integral  
Definition  A definite Integral has upper and lower limits if 'a' and 'b' are the limits or boundaries. The definite integral of f(x) is a number, not function .  An integral without upper limit and lower limit. It is also an antiderivative. The indefinite integral of f(x) is a function not number. 
Expression 

Topics and subtopics of NCERT Grade 12 Maths Chapter7 Integrals
7.1 Introduction
7.2 Integration as an Inverse Process of Differentiation
7.2.1 Geometrical interpretation of indefinite integral
7.2.2 Some properties of indefinite integral
7.2.3 Comparison between differentiation and integration
7.3 Methods of Integration
7.3.1 Integration by substitution
7.3.2 Integration using trigonometric identities
7.4 Integrals of Some Particular Functions
7.5 Integration by Partial Fractions
7.6 Integration by Parts
7.7 Definite Integral
7.7.1 Definite integral as the limit of a sum
7.8 Fundamental Theorem of Calculus
7.8.1Area function
7.8.2 First fundamental theorem of integral calculus
7.8.3 Second fundamental theorem of integral calculus
7.9 Evaluation of Definite Integrals by Substitution
7.10 Some Properties of Definite Integral
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.1
Question:1 Find an anti derivative (or integral) of the following functions by the method of inspection.
Answer:
GIven ;
So, the anti derivative of is a function of x whose derivative is .
Therefore, we have
Or, antiderivative of is .
Question:2 Find an anti derivative (or integral) of the following functions by the method of inspection.
Answer:
GIven ;
So, the antiderivative of is a function of x whose derivative is .
Therefore, we have the anti derivative of is .
Question:3 Find an anti derivative (or integral) of the following functions by the method of inspection.
Answer:
GIven ;
So, the anti derivative of is a function of x whose derivative is .
Therefore, we have the anti derivative of is .
Question:4 Find an anti derivative (or integral) of the following functions by the method of inspection.
Answer:
GIven ;
So, the anti derivative of is a function of x whose derivative is .
Therefore, we have the anti derivative of is .
Question:5 Find an anti derivative (or integral) of the following functions by the method of inspection.
Answer:
GIven ;
So, the anti derivative of is a function of x whose derivative is .
Therefore, we have the anti derivative of is .
Question:6 Find the following integrals
Answer:
Given intergral ;
or , where C is any constant value.
Question:7 Find the following integrals
Answer:
Given intergral ;
or , where C is any constant value.
Question:8 Find the following integrals
Answer:
Given intergral ;
or , where C is any constant value.
Question:9 Find the following integrals intergration of
Answer:
Given intergral ;
or , where C is any constant value.
Question:10 Find the following integrals
Answer:
Given integral ;
or
, where C is any constant value.
Question:11 Find the following integrals intergration of
Answer:
Given intergral ;
or
Or, , where C is any constant value.
Question:12 Find the following integrals
Answer:
Given intergral ;
or
Or, , where C is any constant value.
Question:13 Find the following integrals intergration of
Answer:
Given integral
It can be written as
Taking common out
Now, cancelling out the term from both numerator and denominator.
Splitting the terms inside the brackets
Question:14 Find the following integrals
Answer:
Given intergral ;
or
, where C is any constant value.
Question:15 Find the following integrals
Answer:
Given intergral ;
or
, where C is any constant value.
Question:16 Find the following integrals
Answer:
Given integral ;
splitting the integral as the sum of three integrals
, where C is any constant value.
Question:18 Find the following integrals
Answer:
Given integral ;
Using the integral of trigonometric functions
, where C is any constant value.
Question:19 Find the following integrals intergration of
Answer:
Given integral ;
, where C is any constant value.
Question:20 Find the following integrals
Answer:
Given integral ;
Using antiderivative of trigonometric functions
, where C is any constant value.
Question:21 Choose the correct answer
The anti derivative of equals
Answer:
Given to find the anti derivative or integral of ;
, where C is any constant value.
Hence the correct option is (C).
Question:22 Choose the correct answer The anti derivative of
If such that f (2) = 0. Then f (x) is
Answer:
Given that the anti derivative of
So,
Now, to find the constant C;
we will put the condition given, f (2) = 0
or
Therefore the correct answer is A .
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.2
Question:1 Integrate the functions
Answer:
Given to integrate function,
Let us assume
we get,
now back substituting the value of
as is positive we can write
Question:4 Integrate the functions
Answer:
Given to integrate function,
Let us assume
we get,
Back substituting the value of t we get,
Question:5 Integrate the functions
Answer:
Given to integrate function,
Let us assume
we get,
Now, by back substituting the value of t,
Question:6 Integrate the functions
Answer:
Given to integrate function,
Let us assume
we get,
Now, by back substituting the value of t,
Question:7 Integrate the functions
Answer:
Given function ,
Assume the 19634
Back substituting the value of t in the above equation.
or, , where C is any constant value.
Question:8 Integrate the functions
Answer:
Given function ,
Assume the
Or
, where C is any constant value.
Question:9 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituting the value of t in the above equation,
, where C is any constant value.
Question:10 Integrate the functions
Answer:
Given function ,
Can be written in the form:
Assume the
, where C is any constant value.
Question:11 Integrate the functions , x > 0
Answer:
Given function ,
Assume the so,
, where C is any constant value.
Question:12 Integrate the functions
Answer:
Given function ,
Assume the
, where C is any constant value.
Question:13 Integrate the functions
Answer:
Given function ,
Assume the
, where C is any constant value.
Question:14 Integrate the functions
Answer:
Given function ,
Assume the
, where C is any constant value.
Question:15 Integrate the functions
Answer:
Given function ,
Assume the
Now back substituting the value of t ;
, where C is any constant value.
Question:16 Integrate the functions
Answer:
Given function ,
Assume the
Now back substituting the value of t ;
, where C is any constant value.
Question:17 Integrate the functions
Answer:
Given function ,
Assume the
, where C is any constant value.
Question:19 Integrate the functions
Answer:
Given function ,
Simplifying it by dividing both numerator and denominator by , we obtain
Assume the
Now, back substituting the value of t,
, where C is any constant value.
Question:20 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituting the value of t,
, where C is any constant value.
Question:21 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituting the value of t,
or , where C is any constant value.
Question:22 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:23 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:24 Integrate the functions
Answer:
Given function ,
or simplified as
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:25 Integrate the functions
Answer:
Given function ,
or simplified as
Assume the
Now, back substituted the value of t.
where C is any constant value.
Question:26 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:27 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:28 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:29 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:30 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:31 Integrate the functions
Answer:
Given function ,
Assume the
Now, back substituted the value of t.
, where C is any constant value.
Question:32 Integrate the functions
Answer:
Given function
Assume that
Now solving the assumed integral;
Now, to solve further we will assume
Or,
Now, back substituting the value of t,
Question:33 Integrate the functions
Answer:
Given function
Assume that
Now solving the assumed integral;
Now, to solve further we will assume
Or,
Now, back substituting the value of t,
Question:34 Integrate the functions
Answer:
Given function
Assume that
Now solving the assumed integral;
Multiplying numerator and denominator by ;
Now, to solve further we will assume
Or,
Now, back substituting the value of t,
Question:35 Integrate the functions
Answer:
Given function
Assume that
Now, back substituting the value of t,
Question:36 Integrate the functions
Answer:
Given function
Simplifying to solve easier;
Assume that
Now, back substituting the value of t,
Question:37 Integrate the functions
Answer:
Given function
Assume that
......................(1)
Now to solve further we take
So, from the equation (1), we will get
Now back substitute the value of u,
and then back substituting the value of t,
Question:38 Choose the correct answer
Answer:
Given integral
Taking the denominator
Now differentiating both sides we get
Back substituting the value of t,
Therefore the correct answer is D.
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.3
Question:1 Find the integrals of the functions
Answer:
using the trigonometric identity
we can write the given question as
=
Question:2 Find the integrals of the functions
Answer:
Using identity
, therefore the given integral can be written as
Question:3 Find the integrals of the functions
Answer:
Using identity
Again use the same identity mentioned in the first line
Question:4 Find the integrals of the functions
Answer:
The integral can be written as
Let
Now, replace the value of t, we get;
Question:5 Find the integrals of the functions
Answer:
rewrite the integral as follows
Let
......(replace the value of t as )
Question:6 Find the integrals of the functions
Answer:
Using the formula
we can write the integral as follows
Question:7 Find the integrals of the functions
Answer:
Using identity
we can write the following integral as
=
Question:8 Find the integrals of the functions
Answer:
We know the identities
Using the above relations we can write
Question:9 Find the integrals of the functions
Answer:
The integral is rewritten using trigonometric identities
Question:10 Find the integrals of the functions
Answer:
can be written as follows using trigonometric identities
Therefore,
Question:11 Find the integrals of the functions
Answer:
now using the identity
now using the below two identities
the value
.
the integral of the given function can be written as
Question:12 Find the integrals of the functions
Answer:
Using trigonometric identities we can write the given integral as follows.
Question:13 Find the integrals of the functions
Answer:
We know that,
Using this identity we can rewrite the given integral as
Question:15 Find the integrals of the functions
Answer:
Therefore integration of =
.....................(i)
Let assume
So, that
Now, the equation (i) becomes,
Question:16 Find the integrals of the functions
Answer:
the given question can be rearranged using trigonometric identities
Therefore, the integration of = ...................(i)
Considering only
let
now the final solution is,
Question:17 Find the integrals of the functions
Answer:
now splitting the terms we can write
Therefore, the integration of
Question:18 Find the integrals of the functions
Answer:
The integral of the above equation is
Thus after evaluation, the value of integral is tanx+ c
Question:19 Find the integrals of the functions
Answer:
Let
We can write 1 =
Then, the equation can be written as
put the value of tan = t
So, that
Question:20 Find the integrals of the functions
Answer:
we know that
therefore,
let
Now the given integral can be written as
Question:21 Find the integrals of the functions
Answer:
using the trigonometric identities we can evaluate the following integral as follows
Question:22 Find the integrals of the functions
Answer:
Using the trigonometric identities following integrals can be simplified as follows
Question:23 Choose the correct answer
Answer:
The correct option is (A)
On reducing the above integral becomes
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.4
Question:1 Integrate the functions
Answer:
The given integral can be calculated as follows
Let
, therefore,
Question:2 Integrate the functions
Answer:
let suppose 2x = t
therefore 2dx = dt
.................using formula
Question:4 Integrate the functions
Answer:
Let assume 5x =t,
then 5dx = dt
The above result is obtained using the identity
Question:6 Integrate the functions
Answer:
let
then
using the special identities we can simplify the integral as follows
Question:7 Integrate the functions
Answer:
We can write above eq as
............................................(i)
for let
Now, by using eq (i)
Question:8 Integrate the functions
Answer:
The integration can be down as follows
let
........................using
Question:10 Integrate the functions
Answer:
the above equation can be also written as,
let 1+x = t
then dx = dt
therefore,
Question:11 Integrate the functions
Answer:
this denominator can be written as
Now,
......................................by using the form
Question:12 Integrate the functions
Answer:
the denominator can be also written as,
therefore
Let x+3 = t
then dx =dt
......................................using formula
Question:13 Integrate the functions
Answer:
(x1)(x2) can be also written as
=
=
therefore
let suppose
Now,
.............by using formula
Question:16 Integrate the functions
Answer:
let
By equating the coefficient of x and constant term on each side, we get
A = 1 and B=0
Let
Question:17 Integrate the functions
Answer:
let
By comparing the coefficients and constant term on both sides, we get;
A=1/2 and B=2
then
Question:18 Integrate the functions
Answer:
let
By comparing the coefficients and constants we get the value of A and B
A = and B =
NOW,
...........................(i)
put
Thus
Question:19 Integrate the functions
Answer:
let
By comparing the coefficients and constants on both sides, we get
A =3 and B =34
....................................(i)
Considering
let
Now consider
here the denominator can be also written as
Dr =
Now put the values of and in eq (i)
Question:20 Integrate the functions
Answer:
let
By equating the coefficients and constant term on both sides we get
A = 1/2 and B = 4
(x+2) = 1/2(42x)+4
....................(i)
Considering
let
now,
put the value of and
Question:21 Integrate the functions
Answer:
...........(i)
take
let
considering
putting the values in equation (i)
Question:22 Integrate the functions
Answer:
Let
By comparing the coefficients and constant term, we get;
A = 1/2 and B =4
..............(i)
put
Question:23 Integrate the functions
Answer:
let
On comparing, we get
A =5/2 and B = 7
...........................................(i)
put
Question:24 Choose the correct answer
Answer:
The correct option is (B)
the denominator can be written as
now,
Question:25 Choose the correct answer
Answer:
The following integration can be done as
The correct option is (B)
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.5
Question:1 Integrate the rational functions
Answer:
Given function
Partial function of this function:
Now, equating the coefficients of x and constant term, we obtain
On solving, we get
Question:2 Integrate the rational functions
Answer:
Given function
The partial function of this function:
Now, equating the coefficients of x and constant term, we obtain
On solving, we get
Question:3 Integrate the rational functions
Answer:
Given function
Partial function of this function:
.(1)
Now, substituting respectively in equation (1), we get
That implies
Question:4 Integrate the rational functions
Answer:
Given function
Partial function of this function:
.....(1)
Now, substituting respectively in equation (1), we get
That implies
Question:5 Integrate the rational functions
Answer:
Given function
Partial function of this function:
...........(1)
Now, substituting respectively in equation (1), we get
That implies
Question:6 Integrate the rational functions
Answer:
Given function
Integral is not a proper fraction so,
Therefore, on dividing by , we get
Partial function of this function:
...........(1)
Now, substituting respectively in equation (1), we get
No, substituting in equation (1) we get
Question:7 Integrate the rational functions
Answer:
Given function
Partial function of this function:
Now, equating the coefficients of and the constant term, we get
and
On solving these equations, we get
From equation (1), we get
Now, consider ,
and we will assume
So,
or
Question:8 Integrate the rational functions
Answer:
Given function
Partial function of this function:
Now, putting in the above equation, we get
By equating the coefficients of and constant term, we get
then after solving, we get
Therefore,
Question:9 Integrate the rational functions
Answer:
Given function
can be rewritten as
Partial function of this function:
................(1)
Now, putting in the above equation, we get
By equating the coefficients of and , we get
then after solving, we get
Therefore,
Question:10 Integrate the rational functions
Answer:
Given function
can be rewritten as
The partial function of this function:
Equating the coefficients of , we get
Therefore,
Question:11 Integrate the rational functions
Answer:
Given function
can be rewritten as
The partial function of this function:
Now, substituting the value of respectively in the equation above, we get
Therefore,
Question:12 Integrate the rational functions
Answer:
Given function
As the given integral is not a proper fraction.
So, we divide by , we get
can be rewritten as
....................(1)
Now, substituting in equation (1), we get
Therefore,
Question:13 Integrate the rational functions
Answer:
Given function
can be rewritten as
....................(1)
Now, equating the coefficient of and constant term, we get
, , and
Solving these equations, we get
Therefore,
Question:14 Integrate the rational functions
Answer:
Given function
can be rewritten as
Now, equating the coefficient of and constant term, we get
and ,
Solving these equations, we get
Therefore,
Question:15 Integrate the rational functions
Answer:
Given function
can be rewritten as
The partial fraction of above equation,
Now, equating the coefficient of and constant term, we get
and
and
Solving these equations, we get
Therefore,
Question:16 Integrate the rational functions
[Hint: multiply numerator and denominator by and put ]
Answer:
Given function
Applying Hint multiplying numerator and denominator by and putting
Putting
can be rewritten as
Partial fraction of above equation,
................(1)
Now, substituting in equation (1), we get
Question:17 Integrate the rational functions
Answer:
Given function
Applying the given hint: putting
We get,
Partial fraction of above equation,
................(1)
Now, substituting in equation (1), we get
Back substituting the value of t in the above equation, we get
Question:18 Integrate the rational functions
Answer:
Given function
We can rewrite it as:
Partial fraction of above equation,
Now, equating the coefficients of and constant term, we get
, , ,
After solving these equations, we get
Question:19 Integrate the rational functions
Answer:
Given function
Taking
The partial fraction of above equation,
..............(1)
Now, substituting in equation (1), we get
Question:20 Integrate the rational functions
Answer:
Given function
So, we multiply numerator and denominator by , to obtain
Now, putting
we get,
Taking
Partial fraction of above equation,
..............(1)
Now, substituting in equation (1), we get
Back substituting the value of t,
Question:21 Integrate the rational functions [Hint : Put ]
Answer:
Given function
So, applying the hint: Putting
Then
Partial fraction of above equation,
..............(1)
Now, substituting in equation (1), we get
Now, back substituting the value of t,
Question:22 Choose the correct answer
Answer:
Given integral
Partial fraction of above equation,
..............(1)
Now, substituting in equation (1), we get
Therefore, the correct answer is B.
Question:23 Choose the correct answer
Answer:
Given integral
Partial fraction of above equation,
Now, equating the coefficients of and the constant term, we get
, ,
We have the values,
Therefore, the correct answer is A.
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.6
Question:1 Integrate the functions
Answer:
Given function is
We will use integrate by parts method
Therefore, the answer is
Question:2 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Therefore, the answer is
Question:3 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Again use integration by parts in
Put this value in our equation
we will get,
Therefore, answer is
Question:4 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Therefore, the answer is
Question:5 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Therefore, the answer is
Question:6 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Therefore, the answer is
Question:7 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Now, we need to integrate
Put this value in our equation
Therefore, the answer is
Question:8 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Put this value in our equation
Therefore, the answer is
Question:9 Integrate the functions
Answer:
Given function is
We will use integration by parts method
Now, we need to integrate
Put this value in our equation
Therefore, the answer is
Question:10 Integrate the functions
Answer:
Given function is
we will use integration by parts method
Therefore, answer is
Question:11 Integrate the functions
Answer:
Consider
So, we have then:
After taking as a first function and as second function and integrating by parts, we get
Or,
Question:12 Integrate the functions
Answer:
Consider
So, we have then:
After taking as a first function and as second function and integrating by parts, we get
Question:13 Integrate the functions
Answer:
Consider
So, we have then:
After taking as a first function and as second function and integrating by parts, we get
Question:14 Integrate the functions
Answer:
Consider
So, we have then:
After taking as a first function and as second function and integrating by parts, we get
Question:15 Integrate the functions
Answer:
Consider
So, we have then:
Let us take ....................(1)
Where, and
So,
After taking as a first function and as second function and integrating by parts, we get
....................(2)
After taking as a first function and as second function and integrating by parts, we get
................(3)
Now, using the two equations (2) and (3) in (1) we get,
Question:16 Integrate the functions
Answer:
Let suppose
we know that,
Thus, the solution of the given integral is given by
Question:17 Integrate the functions
Answer:
Let suppose
by rearranging the equation, we get
let
It is known that
therefore the solution of the given integral is
Question:18 Integrate the functions
Answer:
Let
substitute and
let
It is known that
Therefore the solution of the given integral is
Question:19 Integrate the functions
Answer:
It is known that
let
Therefore the required solution of the given above integral is
Question:20 Integrate the functions
Answer:
It is known that
So, By adjusting the given equation, we get
to let
Therefore the required solution of the given integral is
Question:22 Integrate the functions
Answer:
let
Taking as a first function and as a second function, by using by parts method
Question:24 Choose the correct answer
Answer:
we know that,
from above integral
let
thus, the solution of the above integral is
NCERT solutions for class 12 maths chapter 7 Integrals Exercise: 7.7
Question:1 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
Then it is known that,
Therefore,
Question:2 Integrate the functions in Exercises 1 to 9.
Answer:
Given function to integrate
Now we can rewrite as
As we know the integration of this form is
Question:3 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:4 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:5 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:6 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
a
And we know that,
Question:7 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:8 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:9 Integrate the functions in Exercises 1 to 9.
Answer:
Given function ,
So, let us consider the function to be;
And we know that,
Question:10 Choose the correct answer in Exercises 10 to 11.
Answer:
As we know that,
So,
Therefore the correct answer is A.
Question:11 Choose the correct answer in Exercises 10 to 11.
Answer:
Given integral
So, let us consider the function to be;
And we know that,
Therefore the correct answer is D.
NCERT solutions for class 12 maths chapter 7 integralsExercise:7.8
Question:1 Evaluate the following definite integrals as a limit of sums.
Answer:
We know that,
This is how the integral is evaluated using limit of a sum
Question:2 Evaluate the following definite integrals as limit of sums.
Answer:
We know that
let
Here a = 0, b = 5 and
therefore
Question:3 Evaluate the following definite integrals as limit of sums.
Answer:
We know that
here a = 2 and b = 3 , so h = 1/n
Question:4 Evaluate the following definite integrals as limit of sums.
Answer:
Let
for the second part, we already know the general solution of
So, here a = 1 and b = 4
therefore
So,
Question:5 Evaluate the following definite integrals as limit of sums.
Answer:
let
We know that
Here a =1, b = 1 and
therefore h = 2/n
By using sum of n terms of GP ....where a = 1st term and r = ratio
.........using
Question:6 Evaluate the following definite integrals as limit of sums.
Answer:
It is known that,
..........................( )
NCERT solutions for class 12 maths chapter 7 IntegralsExercise:7.9
Question:1 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:2 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:3 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:4 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:5 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:6 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:7 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:8 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:9 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:10 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:11 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:12 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:13 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:14 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
Multiplying by 5 both in numerator and denominator:
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:15 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
Putting which gives,
As, and as .
So, we have now:
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:16 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
So, we can rewrite the integral as;
where . ................(1)
Now, consider
Take numerator
We now equate the coefficients of x and constant term, we get
Now take denominator
Then we have
Then substituting the value of in equation (1), we get
Question:17 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:18 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
can be rewritten as:
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
Question:19 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
can be rewritten as:
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have
or we have
Question:20 Evaluate the definite integrals in Exercises 1 to 20.
Answer:
Given integral:
Consider the integral
can be rewritten as:
So, we have the function of ,
Now, by Second fundamental theorem of calculus, we have