In what ways did the Enlightenment have an integral part in the ideologies of Romanticism? Do any of them relate/differ? (Perhaps mention important texts from each era.)

The ideologies of Romanticism and the ideologies of the Enlightenment are basically one of the most important ideologies in history, literature, art, science, and pretty much all of the other social, economic, political, and cultural aspects of life. The Enlightenment played an integral part in the ideologies of Romanticism, as it basically inspired its creation. Thus, the ideology of Romanticism came as a response to the ideology of the Enlightenment.
The Enlightenment represented everything rational, concise, scientific, philosophical, and secularistic; Romanticism, on the other hand, represented all that is opposite: irrational, emotional, and instinctual—focused on love, nature, passion, and adventure. Essentially, the Enlightenment focused on the capacity of the human mind, while Romanticism focused on matters of the heart.
You can find more information about several important figures and literary works from both eras here.