Take a thick slice of lemon and rub your underarms daily before you go to have bath. Apply some moisturizer after bath as lemon also tends to make skin dry
Add a pinch of turmeric, honey to lemon juice. This will give you something like a paste. Apply this to your dark underarms and leave for about 10 minutes. Now wash off with water.
Mix lime juice with cucumber juice and add a pinch of turmeric to it. Lemon bleaches and removes dead cells, cucumber provides cooling effect and turmeric prevents discoloration. It is actually an ayurvedic remedy .so easy that you can do it everyday without any hassle
Note:You can also simply cut roundels of cucumber and rub your underarms with it.