References for all the Ayaat are below. It is better to read the Ayaat by yourself as well as listening to it.
أَعُوذُ بِاللَّهِ مِنَ الشَّيْطَانِ الرَّجِيمِ
بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ
Surah Al-Inshirah: v. 1
أَلَمْ نَشْرَحْ لَكَ صَدْرَكَ
Have We not expand for you your breast?
Surah Al-An'am: v. 125
فَمَن يُرِدِ اللَّهُ أَن يَهْدِيَهُ يَشْرَحْ صَدْرَهُ لِلْإِسْلَام
And whomsoever Allah wills to guide, He opens his breast to Islam...
At-Tawbah: v. 14
وَيَشْفِ صُدُورَ قَوْمٍ مُّؤْمِنِينَ
...and heal the breasts of a believing people
Surah TaHa: v. 25, 26
رَبِّ اشْرَحْ لِي صَدْرِيََ وَيَسِّرْ لِي أَمْرِي
"O my Lord! Open for me my chest (grant me
self-confidence, contentment, and boldness).
"And ease my task for me
Surah Yunus: v. 62, 63
أَلَا إِنَّ أَوْلِيَاءَ اللَّهِ لَا خَوْفٌ عَلَيْهِمْ وَلَا هُمْ يَحْزَنُونَ
No doubt! [for] the allies of Allah there will be no fear concerning them, nor will they grieve
الَّذِينَ آمَنُوا وَكَانُوا يَتَّقُونَ
Those who believed and used to fear Allah much
Surah Yusuf: v. 86
قَالَ إِنَّمَا أَشْكُو بَثِّي وَحُزْنِي إِلَى اللَّهِ
He said: "I only complain of my grief and sorrow to Allah
Surah At-Tawbah: v. 40
لَا تَحْزَنْ إِنَّ اللَّهَ مَعَنَا
"Be not sad (or afraid), surely Allah is with us."
Surah Yunus: v. 65
وَلَا يَحْزُنكَ قَوْلُهُمْ ۘ إِنَّ الْعِزَّةَ لِلَّهِ جَمِيعًا
And let not their speech grieve you for all power and honour belong to Allah.
Surah An-Nahl: v. 127, 128
وَلَا تَحْزَنْ عَلَيْهِمْ وَلَا تَكُ فِي ضَيْقٍ مِّمَّا يَمْكُرُونََ
إِنَّ اللَّهَ مَعَ الَّذِينَ اتَّقَوا وَّالَّذِينَ هُم مُّحْسِنُونَ
And grieve not over them, and be not distressed
because of what they plot. Truly, Allah is with those
who fear Him, and those who are good-doers.
Surah Al-Ankabut: v. 33
لَا تَخَفْ وَلَا تَحْزَنْ ۖ إِنَّا مُنَجُّوكَ
"Have no fear, and do not grieve! Truly, we shall save you...
Suarh Ale Imran: v. 139
وَلَا تَهِنُوا وَلَا تَحْزَنُوا وَأَنتُمُ الْأَعْلَوْنَ
So do not become weak, nor be sad, and you will be superior (in victory)...
Surah Fussilat: v. 30
لَّا تَخَافُوا وَلَا تَحْزَنُوا وَأَبْشِرُوا
"Fear not, nor grieve! But receive the glad tidings...
Yunus: v. 58
قُلْ بِفَضْلِ اللَّهِ وَبِرَحْمَتِهِ فَبِذَ‌ٰلِكَ فَلْيَفْرَحُوا هُوَ خَيْرٌ مِّمَّا يَجْمَعُونَ
Say: "In the Bounty of Allah, and in His Mercy - therein let them rejoice." That is better than what (the wealth) they amass.
Surah Al-An'am: v. 82
الَّذِينَ آمَنُوا وَلَمْ يَلْبِسُوا إِيمَانَهُم بِظُلْمٍ أُولَـٰئِكَ لَهُمُ الْأَمْنُ
It is those who believe and confuse not their belief with Zulm (wrong i.e. by worshipping others besides Allah), for them there is security...
Surah Quraish: v. 4
الَّذِي أَطْعَمَهُم مِّن جُوعٍ وَآمَنَهُم مِّنْ خَوْفٍ
(He) Who has fed them against hunger, and has made them safe from fear.
Surah Ar-Raad: v. 28
الَّذِينَ آمَنُوا وَتَطْمَئِنُّ قُلُوبُهُم بِذِكْرِ اللَّهِ ۗ أَلَا بِذِكْرِ اللَّهِ تَطْمَئِنُّ الْقُلُوبُ
Those who believed, and whose hearts find rest in the remembrance of Allah: verily, in the remembrance of Allah do hearts find rest.
Surah Anbiya: v. 87, 88
وَذَا النُّونِ إِذ ذَّهَبَ مُغَاضِبًا فَظَنَّ أَن لَّن نَّقْدِرَ عَلَيْهِ فَنَادَىٰ فِي الظُّلُمَاتِ أَن لَّا إِلَـٰهَ إِلَّا أَنتَ سُبْحَانَكَ إِنِّي كُنتُ مِنَ الظَّالِمِينَ
And [mention] the man of the fish, when he went off in anger and thought that We would not decree [anything] upon him. And he called out within the darknesses, "There is no deity except You; exalted are You. Indeed, I have been of the wrongdoers."
فَاسْتَجَبْنَا لَهُ وَنَجَّيْنَاهُ مِنَ الْغَمِّ ۚ وَكَذَ‌ٰلِكَ نُنجِي الْمُؤْمِنِينَ
So We responded to him and saved him from the distress. And thus do We save the believers.
Surah An-Naas: v. 1-5
قُلْ أَعُوذُ بِرَبِّ النَّاسَِ مَلِكِ النَّاسِِ إِلَـٰهِ النَّاسِِ مِن شَرِّ الْوَسْوَاسِ
الْخَنَّاسِِ الَّذِي يُوَسْوِسُ فِي صُدُورِ النَّاسِِ مِنَ الْجِنَّةِ وَالنَّاسِ
Say, "I seek refuge in the Lord of mankind, The Sovereign of mankind. The God of mankind, From the evil of the retreating whisperer; Who whispers [evil] into the breasts of mankind; From among the jinn and mankind."