How Can Divergent Integrals Be Handled with Ramanujan Summation?

[Klaus_Hoffmann]

Are there any method to deal with divergent integrals in the form

\int_{0}^{\infty}dx \frac{x^{3}}{x+1} \int_{0}^{\infty}dx \frac{x}{(x+1)^{1/2}} ?

in the same sense there are methods to give finite results to divergent series as 1+2+3+4+5+6+7+... or 1-4+9-16+25 or similar

 

[HallsofIvy]

WHAT methods give finite results to 1+2+3+4+5+6+7+... and 1-4+9-16+25 ?

 

[JohnDuck]

WHAT methods give finite results to 1+2+3+4+5+6+7+... and 1-4+9-16+25 ?

I've heard the term Ramanujan summation tossed about in regards to this, but I don't really know anything about it.