If you know where to look, there's plenty of information about "re-capping" old motherboards:
https://www.badcaps.net/recapping_overview/
If you have any practical aptitude with a soldering iron, I can recommend this guy's Youtube video series:
View: https://www.youtube.com/watch?v=jaK38BzEpkM
It's easy. Just look at the side of the bulging capacitor, or an
identical capacitor nearby. Read off the information printed on the label.
You don't need to worry about the manufacturer's name. That's not important.
What you do need is the capacitor's value, measured in microFarads (abbreviated to 'uF') and the working Voltage (abbreviated to 'V').
Capacitors come in a range of standard values and voltages. Price is usually a few dollars at most.
Looking at your photo, the dead capacitor might be 100uF 16V, or 47uF 25V, or 220V 10V. It could be something entirely different.
The critical point is to identify the capacitance (uF) and the Voltage (V). Once you've found this information, head over to eBay, Amazon, CraigsList, AliExpress, or your favourite online market place.
Type in "100uF 16V electrolytic capacitor" (substitute your values if different) and see what comes up.
Measure the diameter and height of your dead capacitor and try to match them with the capacitors on sale. Get your credit card out and buy the capacitor(s).
If you have any doubts about identical capacitors fitted nearby, its worth buying enough capacitors to replace them all. When one capacitor dries out and bursts, the others may not be far behind. You don't have to replace every single capacitor on the motherboard.
Take several photos of the bad capacitor from different angles and make a note of the side marked with a stripe (indicating polarity). It's very important to fit the new capacitor the correct way round. Failure to observe correct polarity will result in the new capacitor going fizz bang.
If the stripe on the side of the capacitor has several '-' (minus) symbols printed on it, this marks the negative lead. If the stripe has several '+' symbols on it, this is the positive side. The motherboard may also be marked with useful '+' and '-' symbols to aid correct fitting of the capacitor.
Now comes the tricky part. You need a powerful (minimum 30W) soldering iron with a small tip, a reel of Solder Wick, some old fashioned 60/40 lead/tin solder (not higher melting point lead-free solder) and a bit of luck. I use a 50W temperature controlled workstation with a selection of soldering iron tips.
https://diystadium.com/how-to-use-solder-wick/
Counter-intuitively, apply a small amount of 60/40 solder to each leg of the bad capacitor. This helps to soak up higher melting point lead-free solder with the Solder Wick.
Follow the web guide above and remove as much solder as possible around each leg of the capacitor. Do not overheat the underlying copper tracks/pads, otherwise they will lift off the printed circuit board (not good). As a rule of thumb, anything over 15 seconds direct heat is too long.
Hold the tip of the soldering iron against one of the wire legs of the capacitor until the remaining solder melts (5 to 10 seconds), then "rock" the capacitor body sideways from the opposite side of the motherboard. It should be possible to gently lever up one side of the capacitor by 1mm. Do not apply too much force. Remove the soldering iron tip.
Repeat the procedure, but heat the other leg and "rock" the capacitor in the opposite direction. Both legs of the capacitor should now be slightly shorter on the track side and the capacitor should have risen 1mm above the motherboard.
Repeat these steps carefully, rocking the capacitor back and forth, until both legs clear the top of the board and it drops out. Do not apply the soldering iron to either leg for more than 10 to 15s, to avoid damage to the board. It doesn't matter how hot the bad capacitor gets, but don't burn your fingers.
When the capacitor is removed, clean the remaining solder from the pads on the underside of the board using Solder Wick. Do not press down too hard or for too long (10s max). You really don't want to lift the tracks off the board.
If you have difficulty clearing the holes, use a sewing needle with a very fine point and press it into the centre of the pad from the under (track) side of the board, NOT from the top (comonent) side.
Carefully check the polarity markings on the new caapcitor and compare them with the old capacitor, together with the photos showing how it was fitted and the polarity. Be very careful if the stripe on the bad capcitor indicates -ve and the stripe on the new capacitor is +ve.
Fit the new capacitor
the right way round. Solder the leads, then cut off the excess, leaving no more than 3mm of wire protruding from the motherboard underside. We don't want a short circuit down to chassis.
Clean the excess flux (brown deposit) left over from soldering with IPA (Iso Propyl Alcohol). Leave board to dry. Check joints to make sure you haven't inadvertently "bridged" the two capacitor leads with a large unsightly blob of solder.
Fit the motherboard back in the computer. Connect all boards and cables, then switch on. Keep your face well clear of the new capacitor, in case you've made an elementary mistake and fitted it the wrong way round. Keep your fingers crossed.
If you're lucky, normal function will be restored. Both DIMMs will work. Well done!
I've restored a number of old boards successfully. You have the satisfaction of a job well done and it saves the planet for a few years from another unwanted motherboard, with all the WEEE health implications implied in responsible disposal.
https://europa.eu/youreurope/business/product-requirements/labels-markings/weee-label/index_en.htm
If you have any doubts whatsoever about your soldering skills, or you lack the correct equipment, hand the board over to someone with more experience. It's a 5 to 10 minute job per capacitor, unless you're particularly unlucky or clumsy.