Here's another problem if anyone is interested.

Can you perform the integration? I am curious to see if anyone has a cool method.

Anyone who knows integration knows that e^{\frac{x^{2}}{2}} is not easily integrated.

\int{(x^{2}+1)e^{\frac{x^{2}}{2}}}dx

It appears no one was interested in trying the problem. Not that anyone cares, but I will post my solution.

Parts or substitution do not work too well. Here is a rather obscure method that works well

when you have something of the form p(x)e^{x}

This is known as integration by recognition.

\frac{d}{dx}[p(x)e^{\frac{x^{2}}{2}}]=(x^{2}+1)e^{\frac{x^{2}}{2}}

Let p(x)=ax^{2}+bx+c

Then \frac{d}{dx}(ax^{2}+bx+x)e^{\frac{x^{2}}{2}}

By the product rule:

(2ax+b)(e^{\frac{x^{2}}{2}})-(ax^{2}+bx+c)xe^{\frac{x^{2}}{2}}=

(x^{2}+1)e^{\frac{x^{2}}{2}}

Factor, group like terms and equate coefficients, we get:

a=0, \;\ b=1, \;\ 2a+c=0

\frac{d}{dx}(xe^{\frac{x^{2}}{2}})=(x^{2}+1)\cdot{ e^{\frac{x^{2}}{2}}}

Now integrate both sides and we get:

\fbox{xe^{\frac{x^{2}}{2}}=\int{(x^{2}+1)e^{\frac{ x^{2}}{2}}dx}

Hi gal, just wanted to say that I enjoy these, don't get disheartened that nobody attempted it.

Also, clean out your inbox, I might need to message you in the future :)

It's cleaned. It wasn't full. About half.

Last time I wanted to send you a message it was full! :o