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where can I learn about differential forms, what they represent and how to integrate them.

PS: Where can I read about vectorial spaces and vectorial analysis, especially how to integrate.

I need these to understand my physics course, classical mechanics.

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There are plenty of books about differential forms and each of us will have a personal favourite. If you are starting out, I would suggest maybe having a look at:

Hopefully one of those will have what you need. I am sure there are also plenty of notes you could find online that cover similar material.

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    What do you like about the Weintraub book? – 2017-02-24
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    I think for someone just starting differential forms, it is a gentle guide into the field without being overloaded with new notation. There's plenty of points that irked me about the book though and it is not where I learnt differential forms (though I can safely say, with the fortune of hindsight, that I wouldn't suggest the book I used as suitable for a first foray into the field!) – 2017-02-24
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Try "Mathematical Physics" by Mary L. Boas. It is a very good textbook for learning things from the essential basics.