How to demostrate that Integrate of |t|dt in {0,x} for any x in R is (1/2)x|x|
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Hint: use the derivative form of the fundamental theorem of calculus.
i dont know how
We are given to verify: (1) The derivative form of the fundamental theorem of calculus is: Use this for the left side, and on the right use the product rule and the fact that we have .
I have tried, but I can't see the relation, excuseme
first I wish to use that t^2/2 is equal to |t|, but im not shure, that it is correct, and in the form that you sugest, I think is better, but sorry, I know that I need more practice, or I have a problem in understand a detail.
Let's focus on the left side first. What do you get using the theorem I suggested when you differentiate?
|t|, {0,x}=f(x)
Or i don´t know if must be only |x|=f(x)
Yes, you get |x|. Now, what do you get when you differentiate the right side? edit: Use the definition .
Last edited by MarkFL; November 3rd 2012 at 09:44 PM.
F'(x)=[(x^2)/2|x|]+(|x|/2)
On the right side, you have: Using the product, power and chain rules, we find: Thus, we have shown the given result for the definite integral is valid for all real .
OOOOOOh!!!! It's realy easy, I was near but I didn't know, when I had to use |x| or x^2/2. I realy thank you for your patience and your time. . .
I only sitll have a doubt, what about the condition that x < or = 0?
That condition was an error on my part. x can be any real number, as originally stated in your first post.
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