How to solve this complicated integration?

Question

The PDF of random variable $h$ is $$f_h(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-x^2/2\sigma^2}$$ I want to integrate PDF $f_\rho(x)$ i.e., $$\int_0^{h}f_\rho(x)\ dx$$ where $$f_\rho(x)=\frac{f_h(\sqrt{x/\rho_t})}{\sqrt{\rho_tx}}$$ I know the final answer includes error function. But having difficulty in reaching to final step. Any help in this is highly appreciated.

Answer 1

If $$f_h=\frac{1}{\sqrt{2 \pi \sigma ^2}}e^{-\frac{x^2}{2 \sigma ^2}}\qquad \text{and}\qquad f_\rho=\frac{f_h\left(\sqrt{\frac{x}{\rho }}\right)}{\sqrt{\rho x}}$$ then $$f_\rho=\frac{e^{-\frac{x}{2 \rho \sigma ^2}}}{\sqrt{2 \pi \rho \sigma ^2} \sqrt{ x}}$$ Let $$t^2=\frac{x}{2 \rho \sigma ^2}\implies x=2 \rho \sigma ^2 t^2\implies dx=4 \rho \sigma ^2 t\, dt$$

$$\int f_\rho \,dx=\frac{2 }{\sqrt{\pi }}\int e^{-t^2}\,dt$$