Two Annoying Problems

[rakuenso]

#1

 

the integral from -infinity to +infinity of the function: e^-(x^2)

 

#2

 

the general integral of the function |x^n|

not written as two integrals

 

 

btw i'm looking for an ALGEBRAIC method, not a graphical method

[matt grime]

1 is famous enough to be in any good analysis book:

 

Let I(x) be the integral you require, then

 

[math]I(x)I(y) = \int \int exp(-x^2-y^2)dxdy[/math]

 

make a substitution and do it in spherical polars

 

[math] \int\int e^{-r^2}rdrd\theta[/math]

 

which is some multiple of pi

 

the second: |x| is defined in two parts, so it must be integrated in two parts, unless n is even in which case the module sign can be dropped.