In both "normal" and "slow" the last sound is clearly 'И' ('книгИ'), but this answer is rejected (Duo expects 'книге').
Both answers are grammatically correct in Russian - the first one (ends with 'книги') answers the question 'Where should we write to?' (to our books), the second one (ends with 'книге') answers the question 'Where should we write?' (in our books), but given that the audio clearly sounds 'книгИ', I reported it.
I would say the Russian sentense is incorrect. If something is already written in the book it is "в книге" and if you are putting/writing it there it is "в книгу". Accusative is normally used for the place of destination, which the book in this sentence is - the destination of the writing.
Grammatically in the books is obviously correct but i don't see how it's physically possible to write "in" a book. If I would be small enough to get into a book, I couldn't for sure hold a pen to write while I am insinde. So "into" should be accepted even if it is not the grammatically correct case.