You should know this if studying differential equations. And you would not be asked to solve things like if integration by parts was not pre-requisite to this topic.
Look, the simple fact is that your ability to solve integrals is way below par for attempting to solve differential equations. You MUST go back and review all the basic integration techniques pre-requisite to a differential equations topic. Otherwise you're in for a world of pain.
The trouble now is that from you're probably going to want to know how to make v the subject.
Yeah, I know it seems like I have no idea what I'm doing (I mostly don't), but my teacher assigned this to us and it's a major grade. And yeah, that's really weird about not knowing the integration by parts thing. I flipped through my book and we're not learning that for a couple more chapters.... Ah well.
To get C, you need to know something about where the object is at a particular time. Typically, you'll be told its initial position.
You could assume that at t = 0, s = 0. Note that since downwards is being taken as the negative direction (and k is less than zero) you will have s < 0 for t > 0 .....