Our leading Ulama who have vast knowledge and long experience in Hadeeth believe that Du'a al-Faraj and Du'a al-Nudba are authentic, that is why they read and advise the believers to read these Du'as.
Du'a e Faraj (ALLAHUMMA KUN LIWALIYYAK AL HUJJAT IBN AL HASAN .....) is narrated by the prominent scholar Al-Kulaini. From Mohammad Ibn Easa ibn Obaid through Ali Ibn Ebraheem and all of them are authentic narrators. Mohammad Ibn Easa ibn Obaid is well known narrator from Imam Mohammad al-Jawad (AS).
Du'a e Nudba was narrated by Mohammad ibn al-Mashhadi is his very well known book Al-Mazaar al-Kabeer. He narrated it from Mohammad ibn Abi Qurrah who narrated it from Mohammad Ibn al-Husain al-Bazoofari who is from the teachers of al-Shaikh al-Mofeed, and from the teachers of Ibn al-Ghatha'eri as Ayatullah Khoo'ee mentioned in Mu'jam Rijaal al-Hadeeth.