What is the goal of this integration? Are you comparing the area of that specific peak to that of a (calibration curve of) pure standard?
What is causing the sudden increase in baseline at 1.5' in the chromatogram you provided? Is this also observed when you run a blank (i.e. water injection)? If so, then I'd suggest to move the baseline for integration up to the level of the baseline after 1.5'.
So it depends, I guess: I don't think there is a correct or incorrect way to integrate the peak. It's about what you'd like to know/calculate and whether this can be applied consistently for all your samples.
The chromagram can not be used as-is, so integration can not be obtained from it. You need to adjust your HPLC method first such that the baseline remains flat (or relatively flat) and not jump up after the first peak and change level (That is why you have all of those drop-down lines and why your data is invalid). *The CDS operator's manual should have a detailed section on integration for you to review. From it, you can learn the basic techniques to use.
"Do you think is more correct move the integration up to the level of the baseline after 1.5 also for this case?" No. The data is trying to tell you something is wrong. We need to listen to it and correct the problem first.
If you could provide us with some basic HPLC conditions, then we might be able to offer some advice. Initially, you should ask someone at your school for help (they must have someone in charge of the instrument). Your school is there to assist you and it is easier to do so in person than on these web forums.
In the meantime, here are some comments:
A few common reasons for this sudden baseline shift are: poor equilibration; column sample overload; co-elution resulting in resetting of the baseline due to peak bleed; An injection solution which is too strong or too high in volume relative to the mobile phase (or column). Ideally, your injection solution should be the mobile phase (what are you using?). The injection volume should be less than 3% of the column volume. What are your inj volume and sample concentration? What are your detection settings (sample rate, wavelength/ bandwidth. flow cell volume and pathlength)? What are your column dimensions? What is your flow rate? What is your mobile phase composition? Is it isocratic or a gradient? What is your column volume and K prime?
I don't know why the baseline increased? However, if you did stress study of drugs here, then I saw this before. There are many integration methods. In you case, you might need to have three time sections for the integration. Section one for peak 1, section two for peak 2, section three for the rest peaks (see attached). However, you need to integrate peaks depending on your purposes. There are many different ways.
The solution to the problem is to correct the method and conditions used, not try and manually make multiple adjustments to "fix" individual issues. I have seen chromatograms like these many times and they are usually a sign of a lack of training. A student would not be expected to know how to deal with problems like this on their own. Good method development will remove most of these problems. That is where the troubleshooting should start.