Two points:
1)The possibility of editing, or extracting text from a PDF document depends crucially on how the document was created. If the PDF originated from someone typing words onto a computer, for example a plain text document, a Word document or text copied/pasted from a Web page, then saved as PDF, then all the original text will be retained inside the PDF file and can be extracted by any of the methods mentioned in the replies. That includes manually selecting one or more paragraphs and copy/paste back into a new document.
But if the PDF was created by scanning pieces of paper, like pages of a newspaper or an old book, then the resulting PDF file will basically be a sequence of image files and can only be viewed or printed out as such. Generally, the way to discover which sort a PDF file is will be to open it and try to select lines of text or individual words. If that is not available then the page has been scanned as an image. That is when you would need to resort to OCR software as suggested by Mike175.
2) After that, the idea that a computer can translate a book from Danish to English is rather laughable. Yes you will see Facebook offer to translate someone's post into English (including quite often something that's already English which gives you an idea how accurate it is!) or a Web page that claims you can read it in many other languages - and you may have an idea how 'good' these translations are from google-translated Web pages from (say) German. You will see that when the software doesn't recognize a word it leaves it in the original language. That may not sound so bad - but if the un-recognized word is the main verb of the sentence, for example, the algorithm will then look for another verb to takes its place and often mess up the entire sentence.
Now consider that in that case, you will be asking the software to 'translate' the result of an OCR scan - which itself will quite probably have anything up to a third of words misinterpreted, and if you don't know much Danish you wont be able to make corrections, so you would be submitting something to the translation software that is pretty much nonsense to start with, with predictable (non) results.
If you were going to ask a human to translate, of course the PDF would be all you needed. And accepting no-one would agree to translate a whole book except as a paid job, you might know someone with enough knowledge of the language to take a quick look and tell you 'these pages are about Jan growing up on a farm, this bit is about Mikael's time in the military, there's a long piece about the state of education in the 1820s, that chapter ...' so you have an idea of the content.
