I need to find

$\displaystyle \int_0^{\infty}1-\tanh\left(\frac{x}{a}\right) dx$

which I get to be

$\displaystyle x-a \ln\left(\cosh \left(\frac{x}{a}\right)\right)$

But I'm not sure how to use the limit at infinity for this. I'm also pretty sure I'm not meant to be getting zero at the answer but something depending on a.

Any help would be greatly appreciated!