Sure, it is one of those many techniques which helped fill up the books listing known antiderivative formulas that we used to consult last century, and which were programmed into computer algebra systems like Maple. But like long division, apart from some pedagical value in mathematical thinking for elite math majors, they are no longer relevant to today's real world of using mathematics. It is far more important to understand how to use integrals and what they mean in context than waste precious college calculus time on integration techniques in chapter 7.
BUT integration by parts is important NOT AS AN INTEGRATION TECHNIQUE!
Instead, all of modern day theoretical physics rests on a foundation that requires integration by parts to justify. One learns about this story in an upperclass physics mechanics class. The mathematical name for this topic is "calculus of variations", but don't go googling it, because you probably won't find a discussion that is aimed at freshman or sophomore calculus students.
The classic example of a problem in this topic would be the following. suppose you take a donut surface (called a torus). Take 2 nearby points on the surface. Find the curve in the surface connecting them which has the shortest length. How does one approach this problem? How does one even pose the question mathematically?
You will learn how to find the length of a plane curve in chapter 8. It is simply an integral over the curve whose value depends on the path it takes to connect the two fixed endpoints of the curve, which are assumed to be given. The unknowns are the coordinates of the position functions along the curve, namely functions of a single variable. There are an infinnite number of curves which connect up two (nearby) points, but only one which extremizes the arclength of a curve between them. Without integration by parts, we would not be able to solve this problem. Here is a popular article on this problem.