Your question is Gaussian Integral Evaluation. Take a moment with it on the right.
Talk me through your thinking if you like. When you're confident, submit your answer and I'll grade it like a real screen (7/10 or better passes).
Evaluate this integral, specifically the Gaussian integral.
Evaluate the improper integral
Provide an exact closed-form result and justify convergence. A complete solution should use the standard technique of squaring the integral, converting the resulting double integral to polar coordinates, and evaluating the resulting radial integral. Also explain how the result relates to the normalization constant of the standard normal distribution.