HSC Tips - Integration (1 Viewer)

[freaking_out]

Originally posted by CM_Tutor
My answer remains no, and here's why: There are very few questions 'later on' in an HSC that will invove integration finding an indefinite integral, where you will not need to then find the constant C. Get into the habit of always writing the '+ C'm as you don't know when you're going to need it.

Also, you write in different booklets as they are marked by different people, so no one marking a Q 5 will go back and check if you added the '+ C' in earlier questions.

yeah, of course its better to get in to the habit of writing +C, but say in question 5 (a) u wrote +C and in 5 (b) u didn't, then i think u don't get a mark deducted for that. :S

 

[AGB]

wouldnt it be better to be on the safe side anyway?? i mean, its gonna take, what, an extra two seconds at max?? thats not very much to get an extra mark (possibly) you would not have otherwise gotten...

 

[CM_Tutor]

Originally posted by freaking_out
yeah, of course its better to get in to the habit of writing +C, but say in question 5 (a) u wrote +C and in 5 (b) u didn't, then i think u don't get a mark deducted for that. :S

I don't, either. But then, I don't think questions 5 (a) and (b) will be as easy as finding indefinite integrals.

PS: In my Extn 1 HSC exam, the only mistake I know about outside of the end of q 7 flowed from forgetting to add a + C in a circumstance where I needed to find it...

 

[freaking_out]

Originally posted by AGB
wouldnt it be better to be on the safe side anyway?? i mean, its gonna take, what, an extra two seconds at max?? thats not very much to get an extra mark (possibly) you would not have otherwise gotten...

yeah, thats true but i was just telling u guys what i heard thats all.

 

[Grey Council]

Originally posted by freaking_out
yeah, of course its better to get in to the habit of writing +C, but say in question 5 (a) u wrote +C and in 5 (b) u didn't, then i think u don't get a mark deducted for that. :S

Originally posted by AGB
wouldnt it be better to be on the safe side anyway?? i mean, its gonna take, what, an extra two seconds at max?? thats not very much to get an extra mark (possibly) you would not have otherwise gotten...

hrm, unless i'm mistaken, he meant if you forget to write a + C in the second part.
I mean, to do it on purpose is not very smart. But if you forget, you shouldn't get a mark taken off.
lol, i really have to get into the habit of writing a + C. I tend to sit back and admire the solution, completely forget the + C.

 

[gman03]

From what I heard, for each question, if you forgot at least 1 +C, then you only be deduced with a maximum of 1 mark...

Another tips for integration would be integrating sec x, where you multiply by (sec x + tan x) / (sec x + tan x) .. I read this in another thread by Cm_Tutor

 

[coca cola]

2.)direct substitutions ( f'(g(x)*g'(x) integrals)

What does this: ( f'(g(x)*g'(x) integrals) mean?

 

[withoutaface]

If you have an integral with limits of pi/2 and 0, look for a t=tan(x/2) substitution.

If you have an integral sqrt(a^2-x^2) then substitute in x=asin@

But this is probably pretty useless, since 90% of the time they will give you the substution anyway.

 

[SDuke]

Yo guys use the "LIATE" Technique when doing integration by parts ?
e.g, what to let u', u, v', v

Yeah I do, it's a handy way of doing integration by parts.

LIATE is an acronym for: log (1st priority), inverse function, algebra, trig and exponential (Last priority)

LIATE helps me to remember which part is the u and the v’ of the question; as each letters correspond to the order of priority.

 

[Benny1103]

Another way of integrating secx and cosecx(I'm not sure whether this method or another one is more quick since I haven't checked) is to multiply by (cosx/cosx) and (sinx/sinx) respectively and then use relevant identity + substitution + partial fractions.

 

[niall045]

so for the LIATE technique, does that mean we are assigning this to u?

 

[who_loves_maths]

Originally Posted by Benny1103
Another way of integrating secx and cosecx(I'm not sure whether this method or another one is more quick since I haven't checked) is to multiply by (cosx/cosx) and (sinx/sinx) respectively and then use relevant identity + substitution + partial fractions.

err... plz elaborate on this? it doesn't make sense to say multiply by (cosx/cosx) and (sinx/sinx) and then say to "use relevant identity + substitution + partial fractions." ??? because you are integrating a SPECIFIC integral {ie. secx, or, cosecx}, not a family of similar integrals... so why would you use all that identity + substitution + partial fractions? if you need to use all that, then this is NOT a "quick" way of doing it.
so plz be a little bit more specific?

 

[FinalFantasy]

maybe he means int. sec x dx=int. sec x(secx+tanx)\(secx+tanx) dx
=ln |secx+tanx|+C

 

[who_loves_maths]

^ yea well in that case, it would save ppl having to integrate by parts. but that's not really a "trick", it's taught in most textbooks anyway.

Edit: it's funny cause that's not really an original method that would immediately appear intuitive to ppl. that method only exists because it was initially found that Int[sex dx] = ln(secx +tanx) +c through integration by parts, and after that to make it more elegant and succinct, ppl just multiplied top and bottom by whatever is in the 'ln' brackets... its a method that works for many more intersting integrals other than just 'secx', but i still prefer the more systematic way - shows that you know what's going on, rather than just doing a transformation that's taught to you by a book or a person.

 

[FinalFantasy]

yeap
------------

 

[Benny1103]

Lol its been so long since I've checked this post. Anyway, I was not aware that the multiply denominator and numerator technique(used in the way I presented) is a technique covered in most textbooks. Certainly, I never saw it in a VCE textbook.

Methods may or may not appear obvious to people but as you do more and more questions you get a feel for what is required. For instance how would you integrate the reciprocal of cos(2x) + 1? Use a trig identity, things like that are obvious. Or should I(even in an exam) try to come up with a completely original method to find the integral just in case another student happened to decide to use a trig identity to find the integral?

In the end, as long as a technique is mathematically correct then it is no better or no worse than another, provided that additional requirements are not specified. All techniques and 'transformations' are taught to you by a book or person, whether it is directly or through some other way. Unless you are the kind of person who one day just thought, hey if the integrand of an integral is a product, I could find its antiderivative by reversing something..hmm...that's it...the product rule. I've just deduced two new techniques.

As for what I was suggesting. Say that you wanted to find an antiderivative of sec(x) which is the reciprocal of cos(x). Personally I don't see how multiplying by (sec(x) + tan(x))/(sec(x) + tan(x)) could be more intuitively obvious than doing the same with cos(x)/cos(x). Anyway so multiply the integrand by cos(x)/cos(x).

This will give cos(x)/((cos(x))^2).

(1) Rewrite the square of cos(x) as 1 - (sin(x))^2. That was the first step, I suppose that must have been a very time consuming step.

(2) Let u = sin(x) => du = cos(x)dx. So the integrand becomes 1/(1-u^2). Again, a very time consuming step. Sure to strain the brain a bit.

(3) Use partial fractions or immediately recognise what the partial fraction decomposition is. I'd say that most competent students would be able to do so.
The integrand is now (1/2)((1/(1-u))+(1/(1+u))).

(4) Integrate and you get (1/2)log|(1+u)/(1-u)| + c. Back substitute and you have an answer. Note: log denotes the natural logarithm...standard notation in some places.

None of those steps were all that difficult nor time consuming. On an exam do you think someone who used a totally different and 'original' approach would receive more marks than someone who used an 'unoriginal' technique, even if they both obtained the same answer through mathematically correct arguments?

 

[FinalFantasy]

hey but the more "original" approach to integrating sec x dx ... wouldn't it be
let t=tan (x\2)....

 

[Benny1103]

Well in the end whichever techniques someone wants to use in an exam is his/her loss/gain. I only included my little trick to show a possible way to do the integration.

 

[FinalFantasy]

yea dats cool

 

[MAICHI]

Did anyone mention reduction formulas? You should learn that if you don't already know it. Make sure you know the basic ones: http://archives.math.utk.edu/visual.calculus/4/recursion.2/