BTW - not true that the battery can't be replaced. It can. It just isn't easy like it is with most phones.
Most techies can do it, though the right equipment is needed. If the phone is still under warranty - go to the Apple Store! If not, tools and instructions (and I think the batteries too) can be found at ifixit.com. But - replacement is at least somewhat difficult, and if not done right can mess up the phone, so if you aren't sure about doing it, take it to the Apple store anyway. There will be a charge if it is no longer under warranty.
Having said all that - depending on how old the phone is (I know it's a 4s, but don't know if you got it as soon as it came out, if it's a recent purchase, etc.), what kind of use you put it through, and how "abused" the battery is (leaving it in extreme temperatures, such as a hot car, for example, can really do a number on the battery), the problem may not be the battery at all.
is the best article I've seen on helping with battery life for iOS devices. You don't have to follow everything in the article - if some of the settings are things you REALLY use, keep them. But using the recommendations there to turn off as much "power sucking" as possible should help.
And as someone else said, do make sure to quit your apps once in a while (double-clicking the home button, then flicking the apps up to quit them). I try to quit any apps I'm not actively using - and some are worse about sucking power than others.
A big one: definitely turn off fetch for your mail if at all possible, or at least set it to check a lot less frequently. If fetch is turned off altogether, it will still check for new mail any time you bring up the mail app. Since most people rely on texting for messages they want a quick response to, turning off fetch on mail is usually not a big deal.
If you scroll down on that page, there should be a second article about caring for your battery. That's a good thing to read and follow as much as you can too. You may want to do a couple of full charge/complete drain cycles.
Finally, if you haven't done a full reboot recently, that can help too - especially if you haven't done one since the last time you updated iOS. To do a full reboot, hold down both the power button (top) and the home button (front) at the same time. Keep holding them down until the Apple symbol appears on the screen. Before that happens, you will see the "turn off the device" slider. Ignore it, and keep holding down both buttons at that point. Turning it off with just the power button or with that slider does not do a full reboot of the device.