Help on primitives (1 Viewer)

[Petinga]

Find the primitive of:
a) 2x / (x^2 +1)

b) 3 + (1/x)

 

[Slidey]

a) You should recognise this as: Integral of f'(x)/f(x), which is ln(f(x)) + C
In this case, f(x) is x^2+1, f'(x)=2x, thus the integral is ln(x^2+1) + C

b) The integral of 1/x is ln(x). The integral of any constant, such as 3, is the constant times x, so 3x. The answer is 3x+ln(x)+C.

 

[angmor]

are primitives the opposite of differentiation? so isnt it just a case of working backwards? and i dont think we have actually started intergration yet ;(

 

[insert-username]

The "primitive function" is the anti-derivative, and yes, they are found through reversing differentiation. Integration itself also involves primitive functions, so make sure you learn how to do it.


I_F

 

[angmor]

can u give me a brief introduction to intergration? i hear evry1 talk abt it but my class hasnt even finished primitives yet....what are they used for?

 

[insert-username]

Very briefly, integration is the process of finding the area under a curve - the reverse to differentiation, which is the process of finding the slope of a curve. The basic theory behind integration is similar to differentiation bby first principles - if you draw a rectangle under your curve, you get an approximate area. If you find the total area of all the reactangles as their width goes to 0, you get an accurate answer for the area under the curve.

It sounds complicated, but it works out much nicer than that thanks to all the boffins who've slaved away figuring out shortcut methods for us.


I_F

 

[angmor]

is there a thread on this forum discussing integration? thanks btw....

 

[insert-username]

You'll handle it much better if you work through it with your class. I found integration to be one of those topics which I just needed someone there explaining it - it's much harder to do it across a forum.


I_F

 

[angmor]

yeh i know what you mean...its pretty hard to draw graphs with text only xD....thanks anyways

 

[pLuvia]

The opposite of differentiation. It is how we find out the current formulae for Volume and area of spheres, cones etc.

Integration is one of the easiest topics for me

 

[Riviet]

It is also one of the easiest topics in extension 2 maths. XD

 

[ephemeral]

i've never, ever seen it referred to as the "primitive", in Victoria it's always "find the integral of".

 

[Riviet]

In the hsc exam, they usually don't use the word "primitive", instead they just use a less ambiguous word like "find" or "evaluate" if the integral has boundaries.

 

[sando]

It makes more sense to me when referred to as 'primitive'

 

[SeDaTeD]

The primitive or antiderivative is a function whihc differentiated would yield the original function. An integral is the summation of the f(x).delta_x where delta_x -> 0 (also representing the signed area under the curve). When certain conditions hold, the fundamental theorem of calculus says that the primitive and the integral of a function are the same. (basically, if the function is continuous and differentiable). Thus, you can find the integral by antidifferentiating.

A case where it doesn't hold, for example, find the area under the curve f(x) = 1/x, from x=-1 to x=1. The function is not continuous at x=0, so you cannot apply the fund thm of calculus, ie you can't say the area is INT (from x=-1 to x=1) [dx/x].