The thing about the Psalms is, before their number was definitively fixed at 150 (or rather, even though their number was more or less fixed at 150), there were quite a number of different hymns floating around alongside them, some of them attributed to David. A few of them (Psalms 151-155) at least made it into some versions of Scripture, even if they were not totally regarded as being on the same level as the other psalms (cf. the Septuagintās comment about Psalm 151 being āoutside the numberā).
The 1st century AD
Psalms Scroll (11Q5) found in Qumran, aside from including thirty-nine of the 150 canonical psalms (which are arranged slightly differently than they are today), includes some original material like the Hebrew texts of Psalm 151 (here divided into two psalms), 154, 155, Sirach 51, and a few other previously-unknown hymns. This is how Psalm 151a and 151b goes in there:
Psalm 151A
Hallelujah. Of David, son of Jesse.
1 I was smaller than my brothers, and younger than the sons of my father.
He made me shepherd of his flock, and a ruler over his kids.
2 My hands have made a pipe and my fingers a lyre.
I have rendered glory to the Lord; I have said so in my soul.
3 The mountains do not testify to him, and the hills do not tell (of him).
The trees praise my words and the flocks my deeds.
4 For who can tell and speak of and recount the works of the Lord?
God has seen all, he has heard all, and he listens to all.
5 He sent his prophet to anoint me, Samuel to magnify me.
My brothers went out to meet him, beautiful of figure, beautiful of appearance.
6 They were tall of stature with beautiful hair, yet the Lord did not choose them,
7 He sent and took me from behind the flock, and anointed me with holy oil,
as a prince of his people, and a ruler among the sons of his Covenant.
Psalm 151B
The first display of Davidās power after Godās prophet had anointed him.
1 Then I saw the Philistine taunting [from the enemy lines] ā¦