- #1

- 587

- 243

- Homework Statement
- Can this gaussian integral be solved?

- Relevant Equations
- $$\int^{\frac{x}{\sqrt{t}}}_0e^{-\frac{s^2}{4}}ds+d$$

Solution attempt:

we make the substitution ##\frac{s}{2}=u## and ##ds=2du## to compute

$$\Big(2\int^{\frac{2x}{\sqrt{t}}}_0e^{-u^2}du\Big)^2=2\int^{\frac{2x}{\sqrt{t}}}_0e^{-u^2}du2\int^{\frac{2x}{\sqrt{t}}}_0e^{-v^2}dv=4\int^{\frac{2x}{\sqrt{t}}}_0\int^{\frac{2x}{\sqrt{t}}}_0e^{-(u^2+v^2)}dudv$$we consider the function ##e^{-r^2}## on the plane ##R^2## and compute the integral

$$4\int\int_{[0,{\frac{2x}{\sqrt{t}}}]\times[0,{\frac{2x}{\sqrt{t}}}]}e^{-r^2}rdrd\theta$$

we make the substitution ##\frac{s}{2}=u## and ##ds=2du## to compute

$$\Big(2\int^{\frac{2x}{\sqrt{t}}}_0e^{-u^2}du\Big)^2=2\int^{\frac{2x}{\sqrt{t}}}_0e^{-u^2}du2\int^{\frac{2x}{\sqrt{t}}}_0e^{-v^2}dv=4\int^{\frac{2x}{\sqrt{t}}}_0\int^{\frac{2x}{\sqrt{t}}}_0e^{-(u^2+v^2)}dudv$$we consider the function ##e^{-r^2}## on the plane ##R^2## and compute the integral

$$4\int\int_{[0,{\frac{2x}{\sqrt{t}}}]\times[0,{\frac{2x}{\sqrt{t}}}]}e^{-r^2}rdrd\theta$$