Question

How will you find the antiderivative of \[{{e}^{-2{{x}^{2}}}}\] ?

Answer 1

Hint: To find the antiderivative of a function we use integration. We integrate the function and thus we get the antiderivative. In some cases, we have to use imaginary error functions as those integrations cannot be formed using elementary functions. The above exponential function cannot be expressed into elementary functions so we will use an imaginary error function in order to get the antiderivative.

Complete step by step answer:
The above question belongs to the concept of integrals of the exponential function. Here we have to use the imaginary error function to find the antiderivative. I Imaginary error function is denoted by erf and it is a complex function for a complex variable, here, the complex variable is the non-elementary function in this case we have \[{{e}^{-2{{x}^{2}}}}\] . mostly the error function is used for statistical computation or we can say the standard normal cumulative frequency. This function is very useful in determining the bit error rate of a digital communication system.it is an odd function and when the function reaches the positive side of the infinity the value is one.

In the question we have \[{{e}^{-2{{x}^{2}}}}\] . Here we can’t use elementary functions to find the antiderivative.
Therefore, we will use an imaginary error function.
The definition of error function is \[\dfrac{2}{\sqrt{\pi }}\int{{{e}^{-{{x}^{2}}}}}dx\]
Here we will make a substitution before solving
\[\begin{align}
  & u=\sqrt{2}x \\
 & \Rightarrow du=\sqrt{2}dx \\
 & \Rightarrow \int{{{e}^{-2{{x}^{2}}}}}dx=\dfrac{1}{\sqrt{2}}\int{{{e}^{-{{u}^{2}}}}}du \\
 & \Rightarrow \int{{{e}^{-2{{x}^{2}}}}}dx=\dfrac{1}{\sqrt{2}}\dfrac{\sqrt{\pi }}{2}\text{erfi(-}u)+C \\
 & \Rightarrow \int{{{e}^{-2{{x}^{2}}}}}dx=\dfrac{\sqrt{\pi }}{4}\text{erfi(-}\sqrt{2}x)+C \\
\end{align}\]
Hence, the answer is \[\dfrac{\sqrt{\pi }}{4}\text{erfi(-}\sqrt{2}x)+C\] .

Note:
Antiderivative means the inverse of derivative which is integration. The antiderivative of the given function cannot be calculated unless we have the concept of an imaginary error function. Keep in mind the definition of the imaginary error function for future use. Perform the integration carefully. Do not forget to add the constant of integration.