# Thread: help with this :(

1. ## help with this :(

let $\displaystyle f(x) = \left[ {\int\limits_0^x {e^{ - t^2 } dt} } \right]^2 \,\,\,\,\,\,g(x) = \left[ {\int\limits_0^1 {e^{ - x^2 (1 + t^2 )} } \cdot \frac{1}{{(1 + t^2 )}}} \right]^2$

show that $\displaystyle f'(x) + g'(x) = 0$

and show that $\displaystyle f(x) + g(x) = \frac{\pi }{4}$

finally prove that $\displaystyle \int\limits_0^\infty {e^{ - t^2 } dt} = \frac{{\sqrt \pi }}{2}$

i cant seem able to show that f'(x)+g'(x)=0, i think its beacause i cant derivate g(x). Help plz

2. first $\displaystyle g(x)$ was not meant to be defined as written, it's actually, without the square.

let $\displaystyle h(x,t)=\frac{e^{-x^{2}\left( 1+t^{2} \right)}}{1+t^{2}},$ and $\displaystyle \frac{\partial h}{\partial x}=-2xe^{-x^{2}\left( 1+t^{2} \right)},$ both continuous, then $\displaystyle g'(x)=-2x\int_{0}^{1}{e^{-x^{2}\left( 1+t^{2} \right)}\,dt},$ and $\displaystyle f'(x)=2e^{-x^2}\int_0^x e^{-t^2}\,dt,$ now substitute $\displaystyle t=ux$ in this integral and that shows that $\displaystyle f'(x)+g'(x)=0,$ thus $\displaystyle h(x)=f(x)+g(x)$ is constant and $\displaystyle h(0)=\frac\pi4$ and then $\displaystyle h(x)=\frac\pi4,$ obviously $\displaystyle \lim_{x\to\infty}h(x)=\frac\pi4,$ then $\displaystyle \lim_{x\to\infty}h(x)=\left(\int_0^\infty e^{-t^2}\,dt\right)^2$ and the conclusion follows.

3. A very nice way to find standart integral to be $\displaystyle \frac{\sqrt{\pi}}{2}$.
Could you help to understand that t=ux the sum of derivatives =0.
Thanks.