What to do when mobile gets wet and the battery is not removable I got my mobile (HTC One X) wet yesterday and I could not take out the battery since it is not removable. 

Currently, everything works fine but I can see a dark blur shadow on the right of the screen and few dark spots all over the screen.

What should I supposed to do ? Do I need to take it to technician right away or I can stay calm ?
It looks like your screen has been effected by the water and needs a new screen.

You can also open it up and dry the screen and digitalizer carefully. Maybe the screen and digitalizer are still wet on the inside. iFixit has said that it's "impossible" to repair it (open it up and being able to close it again), so be warned.
The other option is to turn your phone off and put it in the oven for a short few minutes. The heat will come into the phone and the water will dry up, but it's not guarenteed, since it looks like the screen has already been infected.

It's best to let a technician do it's thing. Take note that waterdamage is probably not a warranty thing, so that they're charging serious money. It's your choice to let it repair or not, because I don't think your phone will suddenly stop functioning. If it does, you can always let your phone being repaired later.