Question

I'm working through an example and I get stuck here:
1/(x+sqrt{x^2 +1}) (1 + (2x /(2sqrt{x^2 +1}))
it ends as 1/(sqrt{x^2 +1})
I don't know how to manipulate the second part of the first expression.

Thanks for your insight.

Answer 1

A little hard to read:
\[\frac{1}{x+\sqrt{x^2+1}}\left( 1+\frac{2x}{2\sqrt{x^2+1}} \right)\]

cancel the 2's
for the first fraction, multiply top and bottom by x-sqrt(x^2+1)
this makes the denominator x^2 - (x^2+1) = 1
put the 1 in the 2nd term over a common denominator and add.
if you multiply through the top simplifies to 1