Integral of $$$e^{- x^{2} y^{2}}$$$ with respect to $$$x$$$

Find $$$\int e^{- x^{2} y^{2}}\, dx$$$.

Let $$$u=x \left|{y}\right|$$$.

Then $$$du=\left(x \left|{y}\right|\right)^{\prime }dx = \left|{y}\right| dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{\left|{y}\right|}$$$.

So,

$${\color{red}{\int{e^{- x^{2} y^{2}} d x}}} = {\color{red}{\int{\frac{e^{- u^{2}}}{\left|{y}\right|} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{\left|{y}\right|}$$$ and $$$f{\left(u \right)} = e^{- u^{2}}$$$:

$${\color{red}{\int{\frac{e^{- u^{2}}}{\left|{y}\right|} d u}}} = {\color{red}{\frac{\int{e^{- u^{2}} d u}}{\left|{y}\right|}}}$$

This integral (Error Function) does not have a closed form:

$$\frac{{\color{red}{\int{e^{- u^{2}} d u}}}}{\left|{y}\right|} = \frac{{\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(u \right)}}{2}\right)}}}{\left|{y}\right|}$$

Recall that $$$u=x \left|{y}\right|$$$:

$$\frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{u}} \right)}}{2 \left|{y}\right|} = \frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{x \left|{y}\right|}} \right)}}{2 \left|{y}\right|}$$

Therefore,

$$\int{e^{- x^{2} y^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{y}\right| \right)}}{2 \left|{y}\right|}$$

Add the constant of integration:

$$\int{e^{- x^{2} y^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{y}\right| \right)}}{2 \left|{y}\right|}+C$$

$$$\int e^{- x^{2} y^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{y}\right| \right)}}{2 \left|{y}\right|} + C$$$A