1. ## hard integration

S log(x + (x^2 -1)^1/2)dx

2. Originally Posted by hkdrmark
S log(x + (x^2 -1)^1/2)dx
One approach is to use integration by parts:

$\int u\, dv = uv - \int v \, du$.

Let:

$u = \ln (x + \sqrt{x^2 - 1}) \Rightarrow du = \frac{1}{\sqrt{x^2 - 1}}\,$ (after a modest amount of work),

$dv = dx \Rightarrow v = x$.

For the other five other approaches (where the rest easily follows), await a reply by Krizalid .....

3. that's godly...

4. Originally Posted by mr fantastic
For the other five other approaches (where the rest easily follows), await a reply by Krizalid .....
There's no much here. Integration by parts is the way to tackle integrals like this one.