# Thread: how to calculate this integral!!!

1. ## how to calculate this integral!!!

how to calulate this :

$
\int_{-\infty}^{\infty}e^{-x^2}\ dx
$

it seems easy but i wonder how it is tricky to solve it.
i planned several approaches but all of them failed. i think the answer should be someting similar to $\pi$.

2. Originally Posted by toraj58
how to calulate this :

$
\int_{-\infty}^{\infty}e^{-x^2}\ dx
$

it seems easy but i wonder how it is tricky to solve it.
i planned several approaches but all of them failed. i think the answer should be someting similar to $\pi$.
see post #2 here. you should be able to finish up. it shows you how to find $\int_0^\infty e^{-x^2}~dx$

3. yeah;
that was cool solution.
it was realy nice for me to use double integral trick and then benefit from solving the problem in Polar Coordinates that was undeniably easy.
so the answer is : $\sqrt\pi$

4. Originally Posted by toraj58
yeah;
that was cool solution.
it was realy nice for me to use double integral trick and then benefit from solving the problem in Polar Coordinates that was undeniably easy.
so the answer is : $\sqrt\pi$
yes

5. Originally Posted by toraj58
how to calulate this :

$
\int_{-\infty}^{\infty}e^{-x^2}\ dx
$

it seems easy but i wonder how it is tricky to solve it.
i planned several approaches but all of them failed. i think the answer should be someting similar to $\pi$.
Its also not hard to show that $\Gamma\left(\tfrac{1}{2}\right)=\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}$, where $\Gamma(x)=\int_0^{\infty}e^{-t}t^{x-1}\,dt$

--Chris

6. Originally Posted by Chris L T521
Its also not hard to show that $\Gamma\left(\tfrac{1}{2}\right)=\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}$, where $\Gamma(x)=\int_0^{\infty}e^{-t}t^{x-1}\,dt$

--Chris
And how do you know that $\Gamma(\tfrac 12)=\sqrt{\pi}$ ?

_____________________________________
Here is a few other ways.

1.
Consider $g(x)=\int_0^1 \frac{e^{-(t^2+1)x^2}}{t^2+1} ~dt$

$\bullet$ Find $g'(x)$

Let $f(x)=\int_0^x e^{-u^2} ~du$
$\bullet$ Show that $g'(x)=-2f'(x) f(x)$

$\bullet$ Integrate this latter formula.

$\bullet$ Find an expression for $g(x)$ with respect to f(x).

$\bullet$ Show that $\lim_{x \to \infty} g(x)=0$

$\bullet$ Conclude : $\lim_{x \to +\infty} f(x)=\sqrt{\frac{\pi}{4}}$

Since the integrand is even, $\int_{-\infty}^\infty e^{-t^2} ~dt=2 \int_0^\infty e^{-t^2} ~dt$

---------------------------------------
2.
$\bullet$ Show that $\forall x \in ]-1,+\infty[,~ \ln(1+x) \leq x$

$\bullet$ Deduce that $\left(1-\frac{t^2}{n}\right)^n \leq e^{-t^2} \leq \left(1+\frac{t^2}{n}\right)^n$

$\bullet$ Integrate the previous inequality and make substitutions to get, $\forall n \in \mathbb{N}^*$ :
$\sqrt{n} \int_0^{\pi/2} \sin^{2n+1}(u) ~du \leq \int_0^{\sqrt{n}} e^{-t^2} ~dt \leq \sqrt{n} \int_0^{\pi /2} \sin^{2n-2}(u) ~du$

$\bullet$ Admit the Wallis integral : $\int_0^{\pi/2} \sin^p(u) ~du \to \sqrt{\frac{\pi}{2p}} \quad \text{ when } p \to \infty$
And find the value of $\int_0^\infty e^{-t^2} ~dt$

7. thanks for alternative solutions.