Answer

Verified

450.9k+ views

Hint:It is not possible to find the integration of a non-elementary function by inspection method.

We will first consider the Assertion (A),

In this assertion they have stated that it is not possible to find $\int{{{e}^{-{{x}^{2}}}}}dx$ by inspection method and to verify that we should know the Inspection Method first.

Inspection Method:

Inspection method of integration is the method to find the family of general solutions which differs from each other by a constant value for a particular function such that if we put the limits we will get the value of integral which can be converted into the form of general solution.

This can simply be expressed using mathematical equations as follows,

First we will see the indefinite integration,

$\int{f(x)}dx$ $=$ $F(x)+c$

Integral of a function general Solution of the integral (Antiderivative of f(x)) + Constant

Now in definite integration we can write,

$\int_{a}^{b}{f(x)=F(b)-F(a)}$

And in the Inspection method $F(b)-F(a)$ can be written in the form of$F(x)+c$.

From all of this we can conclude that,

Through the Inspection Method we find the indefinite integration and get a general form of solution.

Now, consider $\int{{{e}^{-{{x}^{2}}}}}dx$, which is the special type of integration in which we won’t get a general solution and it has a particular solution which is given below,

$\int_{-\infty }^{\infty }{{{e}^{-{{x}^{2}}}}=\sqrt{\pi }}$

i.e. $\int{{{e}^{-{{x}^{2}}}}}dx$ can’t take the form of F(x)+c and only gets a value equal to $\sqrt{\pi }$ which can be calculated using the method or concept of Gaussian Quadrature which is not in our syllabus.

And therefore we can write , It is not possible to find $\int{{{e}^{-{{x}^{2}}}}}dx$ by inspection method.’

And therefore the Assertion (A) is True…………………………………………………. (1)

Now let’s move towards the reason (R) which says that the function given in Assertion (A) is not expressible in terms of elementary functions.

As we all know that the elementary functions consist of all basic functions and composite functions. But they have some exceptions which are called the Non-elementary functions.

The function given in Assertion (A) is an Error function which is a type of Non-elementary function.

Therefore we can say that,

Function is not expressible in terms of elementary functions.

Therefore the Reason (R) is True…………………………………………………………… (2)

As we all know that the Inspection Method is only applicable for Elementary Functions. Therefore we can say that It is not possible to find the integration of a non-elementary function by Inspection Method.

And therefore Reason (R) is the correct explanation for the Assertion (A)……………… (3)

From (1) (2) and (3) we can write the answer a , Both A and R are correct; R is the correct explanation of A.

Answer is Option (a).

Note: As it’s a conceptual problem don’t waste your time in finding the integration of the given function as it will consume your very much time and still not going to give the right answer.

We will first consider the Assertion (A),

In this assertion they have stated that it is not possible to find $\int{{{e}^{-{{x}^{2}}}}}dx$ by inspection method and to verify that we should know the Inspection Method first.

Inspection Method:

Inspection method of integration is the method to find the family of general solutions which differs from each other by a constant value for a particular function such that if we put the limits we will get the value of integral which can be converted into the form of general solution.

This can simply be expressed using mathematical equations as follows,

First we will see the indefinite integration,

$\int{f(x)}dx$ $=$ $F(x)+c$

Integral of a function general Solution of the integral (Antiderivative of f(x)) + Constant

Now in definite integration we can write,

$\int_{a}^{b}{f(x)=F(b)-F(a)}$

And in the Inspection method $F(b)-F(a)$ can be written in the form of$F(x)+c$.

From all of this we can conclude that,

Through the Inspection Method we find the indefinite integration and get a general form of solution.

Now, consider $\int{{{e}^{-{{x}^{2}}}}}dx$, which is the special type of integration in which we won’t get a general solution and it has a particular solution which is given below,

$\int_{-\infty }^{\infty }{{{e}^{-{{x}^{2}}}}=\sqrt{\pi }}$

i.e. $\int{{{e}^{-{{x}^{2}}}}}dx$ can’t take the form of F(x)+c and only gets a value equal to $\sqrt{\pi }$ which can be calculated using the method or concept of Gaussian Quadrature which is not in our syllabus.

And therefore we can write , It is not possible to find $\int{{{e}^{-{{x}^{2}}}}}dx$ by inspection method.’

And therefore the Assertion (A) is True…………………………………………………. (1)

Now let’s move towards the reason (R) which says that the function given in Assertion (A) is not expressible in terms of elementary functions.

As we all know that the elementary functions consist of all basic functions and composite functions. But they have some exceptions which are called the Non-elementary functions.

The function given in Assertion (A) is an Error function which is a type of Non-elementary function.

Therefore we can say that,

Function is not expressible in terms of elementary functions.

Therefore the Reason (R) is True…………………………………………………………… (2)

As we all know that the Inspection Method is only applicable for Elementary Functions. Therefore we can say that It is not possible to find the integration of a non-elementary function by Inspection Method.

And therefore Reason (R) is the correct explanation for the Assertion (A)……………… (3)

From (1) (2) and (3) we can write the answer a , Both A and R are correct; R is the correct explanation of A.

Answer is Option (a).

Note: As it’s a conceptual problem don’t waste your time in finding the integration of the given function as it will consume your very much time and still not going to give the right answer.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE