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Somewhere I read that coffee contains carcinogens unless it's filtered, and I think I've noticed that such filtered coffee seems to be weaker. I let it stay for about 5-10min, and it still tastes weaker than coffee made without a coffee filter (paper one of course).
So my question is, how is a filtered coffee different from unfiltered coffee, assuming we're using a paper filter? Mainly I'm looking for a scientific explanation.
If you want stronger coffee (referring specifically to the amount of total dissolved solids), then you really just need a stronger coffee to water ratio. Or just simply add more coffee. Although what I think you are really experiencing is a difference in mouth feel. Using a paper filter with certainly give the coffee clarity. Really fine paper filters (like a chemex filter - being thick and made from soft woods) will evidently filter out very small particles - which results in a very delicate, crisp, and light mouth feel tasting cup of coffee. On the opposite end of the spectrum you have the french press. Which you end up having tiny fine particles left in your end cup, which gives a great amount of body, and a very heavy mouth feel. Both are totally fine cups of coffee, but are totally subjective to your palate.
The thing about extraction is that you really only want to extract about 20% of the coffee you have. If you look at an extraction versus time graphed out, you can see that it represents the curve y=sqrt(x), where extraction continues to increase as time goes on, but at a slower and slower rate. All this means is that by simply letting your coffee brew longer, you are not exactly going to make it any stronger past a certain point. You simply just need to add more coffee. As for the filter - your over all brew strength, or total dissolved solids - should basically be the exact same compared to if you didn't filter it with paper. The main difference is how it's being presented to your mouth. 
I think a good starting place is a 1:16 ratio, meaning 1 parts coffee to 16 parts water. Depending on how that taste to you, just manipulate it until desired. Hopefully this helps out!
There's two points to address.
"Weaker" coffee with paper filters as compared to brewing right in the cup. As far as I understand (although I can't find citation), paper filter does not capture caffeine or aromatic compounds very much. The feeling that filtered coffee is thinner may in your case come from less microgrounds floating in the cup (i.e., from mechanical, rather than chemical, sensation).
I'm not sure what you meant by carcinogenic properties, but you may want to read about cafestol and kahweol, two diterpines that are supposed to have anti-carcinogenic effect. They are also said to raise cholesterol levels (this study). They are removed by paper filters, and if you want to go into details of its levels in various preparations, see this article. It reports that cafestol is lower in darker roasts, and among the non-filtering methods is highest in swedish boiled coffee and lowest in moka pot ("stovetop espresso").
