Integration of exponential functions: $\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$

Question

How I am to solve this integral? I am not able to use any of the methods. $$\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$$

Answer 1

Rewrite $$ \int_0^\infty e^{\Large-\frac{x^2}{y^2}-y^2}\ dx=e^{\Large-y^2}\int_0^\infty e^{\Large-\frac{x^2}{y^2}}\ dx, $$ where the last form integral is Gaussian integral: $$ \int_{0}^\infty e^{-ax^2}\,dx=\dfrac{1}{2}\sqrt{\dfrac{\pi}{a}}. $$ Letting $a=\dfrac1{y^2}$, then $$ \int_0^\infty e^{\Large-\frac{x^2}{y^2}-y^2}\ dx=\dfrac{|y|}{2}\sqrt{\pi}e^{\Large-y^2}. $$