1. ## integrate 1/(sqrt(x^2-4))

Good afternoon,

I managed to work my way through this problem rather quickly, but wound up with an answer that I feel is not correct (at least according to my calculator).

I have:

$\displaystyle \int{\frac{1}{\sqrt{x^2-4}}dx$

I evaluate to:

$\displaystyle \ln\mid \frac{x+\sqrt{x^2-4}}{2}\mid + C$

It should be:

$\displaystyle \ln\mid{x+\sqrt{x^2-4}}\mid + C$

I am picking up the 2 in the denominator via $\displaystyle sec\theta = \frac{x}{2}$ and $\displaystyle tan\theta = \frac{\sqrt{x^2-4}}{2}$.

Can anyone tell me what it is that I'm doing wrong here?

2. Originally Posted by MechEng
Good afternoon,

I managed to work my way through this problem rather quickly, but wound up with an answer that I feel is not correct (at least according to my calculator).

I have:

$\displaystyle \int{\frac{1}{\sqrt{x^2-4}}dx$

I evaluate to:

$\displaystyle \ln\mid \frac{x+\sqrt{x^2-4}}{2}\mid + C$

It should be:

$\displaystyle \ln\mid{x+\sqrt{x^2-4}}\mid + C$

I am picking up the 2 in the denominator via $\displaystyle sec\theta = \frac{x}{2}$ and $\displaystyle tan\theta = \frac{\sqrt{x^2-4}}{2}$.

Can anyone tell me what it is that I'm doing wrong here?
nothing wrong ...

$\displaystyle \displaystyle \ln\left|\frac{x+\sqrt{x^2-4}}{2}\right| + C = \ln\left|x + \sqrt{x^2-4}\right| - \ln{2} + C$

... which only differs by a constant from the "it should be" solution.