This is topic Yo stupidly simple Calculus question in forum Books, Films, Food and Culture at Hatrack River Forum.

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Posted by Blayne Bradley (Member # 8565) on :

okay so if I have f(x)=(x+2)^2

then f'(x) = 2*(x+2)* (dx/dy) x+2 which is 1

so f'(x) = 2x+4

However if I want to find the integral of 2x+4

so: f(x) = 2x+4 (dx)

x^2+4x+C

Which I imagine if you factor it probably equals to (x+2)^2 if you solve for c

x^2+2x+2x+4
x^2+4x+4

Which seems a little wierd, is the answer good enough at the polynomial x^2+4x+C or do I have to get it back to (x+2)^2?

Next if I have (x+2)^2, do I find its integral? Foil it out and do f(x)= x^2+4x+4 (dx)?

In which case it should be (1/3)(x^3) + 2x^2 + 4x + C?

Or is there a reverse chain rule I can use?

(1/3)[(x+2)^3](x^2+2x)+C?

//one third times x+2 cubed times the intergral of the inside plus a constant?

Posted by Phanto (Member # 5897) on :

Your question doesn't really make sense to me.

quote:

is the answer good enough at the polynomial x^2+4x+C or do I have to get it back to (x+2)^2?

The integral of 2x + 4 is x^2 + 4x + C... where are you getting the whole weird (x+2)^2 stuff from?

The fact that you can rewrite an equation in another way doesn't mean you should, and unless you know what C is for some reason then it is unknown.

Posted by Blayne Bradley (Member # 8565) on :

The point is say I have f(x) = (x+2)^2

The idea is figuring out how to integrate it and how to integrate back from its derivative.

Posted by fugu13 (Member # 2859) on :

You can't, perfectly. Information is lost. Taking the derivative of any constant is 0, so it is impossible to integrate and work out any constant terms that were part of the original.

(x+2)^2 has a constant term -- writing it another way, as x^2 + 4x + 4, you can see that clearly. After taking the derivative, all information about that constant term is lost. It was a 4, but from the derivative you can't tell if it was a 23 or a 7 or 2.1 billion. So no, you can't get back to (x+2)^2 from knowing the derivative is 2x + 4. Because it could have been x^2 + 4x + 3.14159 instead of x^2 + 4x + 4.

Now, if you have additional information, you can compute the constant factor. For instance, sometimes you are given the intercept or somesuch.

Posted by The Rabbit (Member # 671) on :

quote:
okay so if I have f(x)=(x+2)^2

then f'(x) = 2*(x+2)* (dx/dy) x+2 which is 1

so f'(x) = 2x+4

You've lost me completely.

What is y? What variable are you differentiating with respect to?

what is 1?

Aside from that, the answer fugu and Phanto have given you is accurate.

You loose information when you take a derivative, that's why you have to add in an arbitrary constant when you integrate. In order to recover the original function by integrating a derivative, you have to have more information. Typically, you have to know the value of the initial function at some point.

Posted by The White Whale (Member # 6594) on :

I think he means f'(x) = 2*(x+2)*d/dx(x+2), and that d/dx(x+2) = 1.

Posted by Mike (Member # 55) on :

quote:
You loose information...
Speaking of which, I can't wait 'til my information bow arrives in the mail. Posted by willthesane (Member # 11754) on :

the integral from a to b of f(g(x))*g'(x) is the same as the integral from g(a) to g(b) of f(x).

so, the integral of f(x)=(x+2)^2 dx from 0 to 2 is the same as the integral of f(u)=u^2 from 2 to 4.

http://en.wikipedia.org/wiki/Integration_by_substitution

if this is what you were looking for i wish you luck.