Why must the second term on the right-hand side vanish in string theory?

[StenEdeback]

Homework Statement

There is one statement in the text of the attached picture that I do not understand

Relevant Equations

See text below

I am doing private studies in string theory and am reading "A first course in string theory" by Barton Zwiebach. Below equation 6.52 the author
says "Since the second term on the right-hand side must vanish...". I do not understand why this term must vanish, and I would be grateful for an explanation.

Sten Edebäck

 

[MathematicalPhysicist]

It's because of the principle of least action, where $S$ is the action, its variation should be zero.
http://www.scholarpedia.org/article/Principle_of_least_action

 

[MathematicalPhysicist]

The first term on the RHS should vanish because of the choice that boundary values of integration vanish, and the second term on the RHS is zero follows from this and $\delta S=0$.

 

[StenEdeback]

Thank you! A good explanation! Physics Forums is indeed a valuable last resort for me doing private studies, when I cannot find the answers to my questions by googling. Physics Forums is the equivalent of a supporting professor to me. And I feel a true joy when I overcome a hurdle and can go on with my studies. Theoretical Physics is really fun!