There was no need to change the bounds, as long as we integrate with respect to x.
However if you need to use substitution along with by parts, then yes change limits of integration accordingly.
Repeated use of integration by parts
The tabular method to repeated integration by parts is popularized as the “DI Method” by @blackpenredpen (yayyyy).
Let’s look at a (modified version of a) May 2011 question and see how the tabular method is viable.
Evaluate
∫0πe−xsinxdx
It does not really matter which one to differentiate (u) or integrate (v), as both repeats. Differentiate the D column; integrate the I column. The sign alternates, because of the negative sign in integration by parts.
sign
D
I
+
e−x
sinx
−
−e−x
−cosx
+
e−x
−sinx
The key thing is to remember is that diagonal down is uv, and each row is vdu. Let I be the indefinite integral. Associate the sign with D.
The first step is
I=+e−x(−cosx)−∫(−e−x)(−cosx)dx
This gets you a mark. Then write the new diagonal down, as well as the across.
I=+e−x(−cosx)−(−e−x)(−sinx)+∫e−x(−sinx)dx
This gets you another mark. We have found an integral that’s same as the original question (and off by a minus sign), so we stop. Collect the like terms and simplify. Technically when there is only a single indefinite integral in an equation we need the +C. IB generally tolerates missing intermediate +C unless it changes the algebra, such as when solving a differential equation.
The verdict is that as long as you correctly write the results after each use of integration by parts, you will get marks. You just cannot directly go to the answer.
Strategies
If everything looks tough to integrate, use that as the v and use du=1dx,u=x.
Tabular (DI) does not necessarily work for repeated integration if u and v change. See the integration of