Assertion (A) It is not possible to find $\int{{{e}^{-{{x}^{2}}}}}dx$ by inspection method.
Reason (R) Function is not expressible in terms of elementary functions.
a.Both A and R are correct; R is the correct explanation of A.
b.Both A and R are correct; R is not the correct explanation of A.
c. is correct; R is incorrect.
d.is correct; A is incorrect.

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.