the battery of my macbook pro has been expanding severely. the trackpad does not work anymore.. it is a 2011 model. any chance Apple would replace the battery free of charge?

I agree that you should take the expanded battery out and arrange to have it recycled.

In a desktop computer, the DC power supply contains large capacitors to smooth out the DC power supplied to the chips. In a notebook computer, that job is performed by the battery. You can certainly bench test the computer with no battery, and it will operate that way.

it is not prudent to run a Notebook computer without a battery for day-to-day use. No battery means no buffering of DC power, which may leave your notebook computer exposed to damage from power fluctuations.

Sorry.

As this model is a vintage product, Apple will not provide service.

https://support.apple.com/en-us/HT201624

The battery is a wear item, like the soles of your shoes or the brake pads and tires on your car. Yours is beyond its expected life.

Please do not use it with an expanded battery - if left too long, it can explode. At least take out the battery for now and just use with the power cord.