Hyperbolic substitition question:

[Zeth]

Homework Statement

$$\int \!\sqrt {1+{v}^{2}}{dv}$$

Homework Equations

Maple tells me that I have to throw in an arcsinh into the solution some how.

The Attempt at a Solution

I've tried substituting with tan(x) but that got me no where and from the solution I'm given:

$$1/2\,v\sqrt {1+{v}^{2}}+1/2\,{\it arcsinh} \left( v \right)$$

I'm not sure how you get the first term and I know that arcsinh(v) is the integral of 1/sqrt(1+v^2)

 

[matt grime]

If the answer involves inverse sinh, why not put sinh (or cosh or tanh) into the equation?

 

[Gib Z]

If you don't exactly like arcsinh, just rewrite sinh in terms of the exponential function and find its inverse. This shows you that arcsinh is just a fancier way of writing: $$\log_e (x + \sqrt{x^2+1})$$

Edit: P.S. u= tan x was a good idea :) Go along with it.