I want to use amazon alexa api for my website/ mobile application. I would like to know if it is possible to use and how to integrate alexa, given that I develop my custom skill.
You can use the Alexa Voice Service. You will need to collect the user's audio and then submit it to Amazon. The voice snippet does not need to contain the hotword (e.g. Alexa). You should be able to use most features that don't involve the device and aren't asynchronous. For example, you can utilize the alarm or timer features nor the music streams.