Jeg vil tage denne bog med til biblioteket.

Because vil is a modal verb. In Danish, modal verbs are followed by the infinitive of the main verb without at.

So:

• Jeg vil tage denne bog med til biblioteket.
• literally: I will / want to take this book along to the library.

Other similar examples:

• Jeg kan læse. = I can read.
• Hun skal gå. = She has to go / she will go.
• Vi må vente. = We must wait.

So after vil, you use tage, not a conjugated form like tager.

Vil can mean either will or want to, depending on context.

In this sentence, it can often be understood as:

• I want to take this book to the library or
• I will take this book to the library

Danish uses vil more broadly than English will in some contexts. If you want to be very clearly future-oriented, context usually does the job, because Danish often uses the present tense for future meaning too.

Compare:

• Jeg vil tage bogen med. = I want to / will take the book along.
• Jeg tager bogen med i morgen. = I’m taking the book along tomorrow.

So vil does not always map neatly onto just one English word.

Because tage med is a verb phrase meaning take along / bring with you, and in Danish these small particles often appear later in the sentence.

So the full idea is:

• tage med = take along, bring

In the sentence:

• Jeg vil tage denne bog med til biblioteket.

the object denne bog comes between tage and med.

This is very common in Danish:

• Jeg tager min jakke med. = I’m taking my jacket along.
• Hun vil have børnene med. = She wants to bring the children along.

So don’t think of med here as completely independent; it belongs with tage.

Denne means this.

So:

• denne bog = this book

By contrast:

• den bog usually means that book in many contexts, or it can sound like the book with extra emphasis depending on the situation.

For a learner, the safest rule is:

• denne = this
• den = that / it / the one

In modern spoken Danish, many people often prefer:

• den her bog = this book
• den der bog = that book

So all of these are possible, but they are a bit different in style:

• denne bog = correct, somewhat formal or written
• den her bog = very common in speech

Because Danish usually puts the definite article at the end of the noun.

So:

• et bibliotek = a library
• biblioteket = the library

This is one of the big differences from English.

Examples:

• en bog = a book

• bogen = the book

• et hus = a house
• huset = the house

You use det bibliotek only in special cases, for example when adding extra description:

• det store bibliotek = the big library

So in your sentence, til biblioteket simply means to the library.

Because til means to and shows movement toward a place.

• til biblioteket = to the library

By contrast:

• i biblioteket = in the library

So:

• Jeg går til biblioteket. = I’m going to the library.
• Jeg er i biblioteket. = I’m in the library.

In your sentence, the book is being taken to the library, so til is the natural choice.

There is some flexibility, but not unlimited flexibility.

The original sentence:

• Jeg vil tage denne bog med til biblioteket.

is natural and clear.

You may also hear or see:

• Jeg vil tage denne bog til biblioteket.

This is also possible, but it loses the specific sense of taking it along that med gives.

In many cases, med stays after the object:

• Jeg tager bogen med.
• Hun vil tage sin søn med.

Danish word order is shaped by a few rules:

1. In main clauses, the finite verb usually comes in second position.
2. The infinitive stays with its complements later in the sentence.
3. Particles like med often come after the object.

So while Danish does allow some variation, your sentence is already in a very normal order.

You usually put ikke after the modal verb phrase in this kind of sentence:

• Jeg vil ikke tage denne bog med til biblioteket. = I do not want to / will not take this book to the library.

A useful pattern is:

• subject + finite verb + ikke + infinitive

Examples:

• Jeg vil ikke gå. = I don’t want to / won’t go.
• Hun kan ikke komme. = She cannot come.

So ikke normally goes after vil, not at the very end.

Yes, absolutely.

• Jeg vil tage den her bog med til biblioteket.

This is very common in spoken Danish and often sounds more natural in everyday conversation than denne bog.

The difference is mostly style:

• denne bog = more formal, more written, sometimes more careful
• den her bog = common spoken Danish

So both are correct, but den her bog is probably what you will hear more often in daily speech.

It can match either take along or bring, depending on the point of view.

Danish often uses tage med where English might choose either:

• take
• bring
• take along
• bring along

So:

• Jeg vil tage denne bog med til biblioteket.

can correspond to English ideas like:

• I want to take this book to the library.
• I’ll bring this book to the library.

The exact English translation depends on perspective, but the Danish structure itself is normal.

A learner-friendly approximation is:

• biblioteket ≈ bib-lio-TEH-gehth

A few important points:

1. The stress is on -tek-:

   • bib-lio-TEK-et

2. The final -et is the neuter definite ending:

   • bibliotek = library
   • biblioteket = the library

3. In natural speech, the final syllables may sound softer or less sharply pronounced than an English speaker expects.

If you want to sound natural, it helps to focus on the rhythm more than on saying every letter very clearly.

Then Danish main-clause word order changes because the finite verb still has to stay in second position.

So:

• Denne bog vil jeg tage med til biblioteket.

This means:

• This book, I want to take to the library.

Notice what happened:

• first position: Denne bog
• second position: vil
• then the subject: jeg

This is a very important Danish rule called V2 word order: the finite verb comes second in main clauses, even when something other than the subject comes first.