Question

How do you find the integral `int_0^1x*e^(-x^2)dx` ?

Answer 1

We begin by making a substitution.

Let `u=-x^2`

`intxe^udx`

`du=-2xdx`

`(du)/(-2)=(-2xdx)/(-2)`

`(du)/(-2)=xdx`

`(-1)/2*du=xdx`

`inte^uxdx`, Notice that xdx can be replaced by `(-1)/2*du`

`inte^u(-1)/2du`, Move the constant to the front of the integral

`(-1)/2inte^udu`

After integration look back to the original substitution to find the value for `u`. In this case that value is `-x^2`.

I switched back to `-x^2` so that I could use the original boundaries.

`(-1)/2[e^(-x^2)]_0^1=(-1)/2[e^(-1)-e^0]=(-1)/2[1/e-1]=(-1)/(2e)+1/2=`

`=0.31606`