I think you need to consider the context of the situation. The concepts you're talking about are outside the scope of the course she is teaching. Maybe she got defensive, but it probably came off like you were challenging her mathematical competency. I'm not even sure if you're referencing her pride or your own as the the issue. Surely you don't really care if this random teacher things you're wrong about integrals, right?

Remember that your main role here is to supplement the student's learning. Certainly arguing with the teacher over a practice problem that was meant to reinforce topic discussed in class isn't going to do that. For this student to be successful in this class, the need to master the concepts the teacher is covering.

I used to tutor in the 'math lab' when I taught high school, so that meant I would get students from pre-algebra to calc BC. Often the first thing I would have the students do when working on a problem was have them show me a similar problem in their notes. This A) Allowed me to recognize the method they were covering in class and B) Emphasized the importance of good note taking to the student. It helped that I was more familiar with the curriculum, but if a student had just learned the Pythagorean Theorem and they were asked to find the third side of a triangle didn't have a right angle, that was my cue to discuss the necessity 90 degrees for the theorem. That was not my cue to tell them the teacher was wrong for saying the third side couldn't be found since the Law of Cosines exists.