Thread: defined integral

How to demostrate that Integrate of |t|dt in {0,x} for any x in R is (1/2)x|x|

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 .

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.