Question

Evaluate the Intergal

Answer 1

\[\int\limits_{0}^{\infty}e^{-x^{2}} dx\]

Answer 2

Hint: Yes, you CAN do this integral :) .

Answer 3

Hint : Even you can. :)

Answer 4

This is a challenge haha. I already have the solution.

Answer 5

Then keep it with you only.. :)

Answer 6

\[\bigg(\int_0^\infty e^{-x^2}~dx\bigg)^2 =\int_0^\infty e^{-x^2}~dx~\int_0^\infty e^{-x^2}~dx\\
\bigg(\int_0^\infty e^{-x^2}~dx\bigg)^2 =\int_0^\infty e^{-x^2}~dx~\int_0^\infty e^{-y^2}~dy\\
\bigg(\int_0^\infty e^{-x^2}~dx\bigg)^2 =\int_0^\infty\int_0^\infty e^{-(x^2+y^2)}~dx~dy\]
Convert to polar coordinates:
\[\bigg(\int_0^\infty e^{-x^2}~dx\bigg)^2 =\int_0^{2\pi}\int_0^\infty e^{-r^2}~r~dr~d\theta\]
Then take the square root:
\[\int_0^\infty e^{-x^2}~dx = \sqrt{\int_0^{2\pi}\int_0^\infty e^{-r^2}~r~dr~d\theta}\]

Answer 7

Brilliant :) .