I can't answer your original question "how to correctly integrate the given data and also how to set the integration constant?" but I think I know what the problem is.
I was able to read and plot your data. The sinewave plotted is negative (except for the peaks which are zero). Integrating the curve calculates the area between the curve and the x-axis. Since your data is always negative, the area is is increasingly negative , thus the squiggly line trailing down. If you calculate and subtract the mean of your sinewave, i.e.. shift its baseline up to zero and then integrate, the result is more like what you expect -cosine.
It's the joy of real world numerical integration versus theoretical mathematical integration. Unfortunately I have to use differential sensors on regular basis, the output of which has to be integrated to be meaningful. I have yet to find a way to perform the integration reliably. Most of my data is impulses so I know approximately what the initial and final condition should be and adjust the baseline accordingly. Scale is a different matter. Numerical integrations will need a scale factor. Once of my sensors includes the scale factor, but most are homemade and the scale factors are derived my measuring known signals and adjusting accordingly.
A couple of comments on your VI. For whatever reasons, most likely regional settings, I had to use
johnsold format string or every thing was zero.
Also, you are plotting your Time and Y data against the x-axis, which is samples not time.
