Integration generating logs (1 Viewer)

[cwag]

Going through past papers i come accross many integrals that generate an absolute value of a log, or sometimes just the normal parenthesis. ()

e.g ∫ 3.dx/(x-1) = 3 log_e|x-1| + C

but ∫ 2x.dx/(x²+4) = log_e (x²+4) + C

I am just wondering, in what conditions do we generate an absolute value...Would it be when the function is not defined for all real x? like in the second example, there is an x² in denominator, the function would exist for all values of x, so theres no need for absolute values. But in the first example the domain would noramlly be > 1 so we need to define it as the absolute value? I dunno, im probably not making sense...Please help? Thankyou

 

[tommykins]

Going through past papers i come accross many integrals that generate an absolute value of a log, or sometimes just the normal parenthesis. ()

e.g ∫ 3.dx/(x-1) = 3 log_e|x-1| + C

but ∫ 2x.dx/(x²+4) = log_e (x²+4) + C

I am just wondering, in what conditions do we generate an absolute value...Would it be when the function is not defined for all real x? like in the second example, there is an x² in denominator, the function would exist for all values of x, so theres no need for absolute values. But in the first example the domain would noramlly be > 1 so we need to define it as the absolute value? I dunno, im probably not making sense...Please help? Thankyou

Technically you should have absolute values for all the log integrals as log is indefined for x < 0.

|x²| won't matter as it's x², but |x| =/= all x, so theres abit of a problem there.

I'm into the habit of using abosolute values already, I believe it's not too hard to pick up and they can't mark you down for it.

 

[cwag]

so just use absolute value for every integral?....fair enough

 

[vds700]

ive never got into the habit of putting abs values in, better start doing it i guess. Never been penalised for it in my school assessments, but maybe HSC is more strict.

 

[YannY]

ABS value makes the included term positive and x^2+4 cannot be negative hence it doesnt need ABS value.

x^2-1 needs it because when x=0 the term equates to -1 which needs ABS value to change it to positive 1. However if you give |x^2-1| you can say ABS value for -1<x<1 but this is unneccessary.
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[tommykins]

absolute signs always

here (x^2 + 4) = |x^2 + 4| so it doesnt matter

Would it be a good idea just to include absolute values for good measure and technical correctness?

 

[tommykins]

haha, i love the mathematical finesse but its really up to you. i personally include absolute signs in black ink (gel for math work), while in blue i only include them when i have to as you dont get that same sharp definitive contrast with the page. seriously though, gelled ink equations look so much more amazing than in classical biro, and allow for a much more comfortable flow. whatever you do, just dont do math in pencil it looks horrible

i would however imagine a slight statistical advantage of always including absolute signs regardless though, so i would advise to do so

Mathematics in felt tip isn't too bad either, i cringe at pencil though

 

[Trebla]

For the content within the HSC course, it doesn't really matter.

You won't get marked down for not putting the absolute value. Either way is fine.

That being said, it's probably a good idea not to use absolute values in your logs anyway, especially if you look at the integrals across discontinuities it may be misleading if you’re not careful, because even though absolute value is fine technically, it may not assist in helping you realise there is a discontinuity in the problem (assuming you were oblivious about it and did not notice it before i.e. in those moments of making silly mistakes lol). Take the following example: (which is a bit beyond the HSC course)
e.g. ∫_-2¹1/x dx
If you did not realise this is an area across a discontinuity, then you may have blindly done the following:
∫_-2¹1/x dx = [ln |x|]_-2¹
= ln |1| - ln |-2|
= ln 1 - ln 2
= - ln 2
This answer is invalid because if you observe say ∫_-1^a1/x for -1 < a < 0, you get ln (-a) [both with or without absolute values], and as a --> 0, then ln (-a), approaches negative infinity which is undefined.
If you did not realise there was a discontinuity problem, then the absolute value would lead to a misleading answer. If the absolute value was not there, then you would be able to easily recognise that the integral is not well defined which is indicated by the negative number in the log.

As an aside, an interesting example is ∫_-1²1/x² dx
If one does not acknowledge the discontinuity at x = 0, then the result would be:
∫_-1²1/x² dx = [- 1/x] _-1² = - ½ - 1 = - 3/2
Which is ludicrous, as the graph of y = 1/ x² is always positive in the natural domain and any proper integral should give a positive number!

The reason some people may use absolute values is where they encounter something like: ∫_-2^-11/x dx which is in the HSC course.
But if you actually combine the logs straight away before doing any further operations, the problem is eliminated as the negatives cancel each other out to become positive numbers.
∫_-2^-11/x dx = [ln x]_-2^-1
= ln (-1/-2) [rather than: ln (-1) - ln (-2)]
= ln (1/2)
What some people tend to do is:
∫_-2^-11/x dx = [ln |x|]_-2^-1
= ln |-1| - ln |-2|
= ln 1 - ln 2
= - ln 2
= ln (1/2)

So, there’s nothing wrong with using absolute values. Just be aware, it may make way for susceptibility for errors, though such problems probably won't appear in the HSC course.

 

[Harkaraj]

Nice explanation Trebla, I was always unsure of whats up with the absolute values. It seems more clarified now.

 

[Js^-1]

In our trial 4 unit exam, we had a definate integral that resulted in one of the limits of integration being negative. Thats the only time I have used the absolute values for logs.