These are the instructions to root Samsung Galaxy Mega 5.8 GT-I9150 (running on android 4.2.2) using TowelRoot application on Phone itself (no need of computer to root).
Requirements: Your Samsung Galaxy Mega 5.8 GT-I9150 should have at least 10-20 percent of battery to perform the rooting process.
Step 1: Enable installation of 3 party Apps on your Samsung GT-I9150. To Enable, Open Settings > Security > Device Administration > Unknown Sources (tick to enable it).
Step 2: Download and install TowelRoot Application on your Samsung GT-I9150. Once installation is completed you will be able to see the TowelRoot Icon in the App Menu.
Step 3: Now, Tap on the TowelRoot App Icon to Open it.
Step 4: Once TowelRoot Application is launched you will be able to see the following screen on your Device:
Step 5: In TowelRoot, You will be able to see “make it rain” button. Tap on the “make it rain” Option to root your device.
Step 6: Now, you will be able to see the TowelRoot success Message on the Samsung GT-I9150 Screen.
Step 7: Now your Samsung Galaxy Mega 5.8 GT-I9150 is Rooted. To check whether your phone is rooted correctly, open Applications Menu, there you will be able to see a new app installed, called SuperSU. If this app exist then it means you have successfully rooted your device.
Optional: You can also verify the root status of your device by using the Root Checker Application.
[*] No data loss process: Your Samsung Galaxy Mega 5.8 GT-I9150 Data like contacts, pictures, videos, music and etc will be not affected with this method.
[*] Device Warranty: Once you full unroot your device through SuperSU, you can claim for the phone warranty. In short no need to worry for Samsung Galaxy Mega 5.8 GT-I9150 warranty.
[*] How to unroot: To unroot, Open SuperSU > Settings > Full Unroot.
[*] Samsung Stock Firmware: If you are looking for the original firmware for your device then head over to Download Samsung Stock Firmware for all devices page.
[*] Samsung USB Driver: If you are looking for the original USB Driver for your device then head over to Download Samsung USB Driver for all devices page.
[*] Credits: TowelRoot Application is developed by GeoHot (XDA Developer).