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 Posted: Tue May 15, 2007 11:33 am
So, I've got a futher maths test in five weeks for AS level. Now, just to bring a question out into the open- to someone who hasn't done a lot of trigonomic integration/differentiation, how would you recognise and remember what rules to use in different instances? How did any mathematicans remember them? I know the results for integrating different trig/general functions, but if anyone could share how they remember indicators, I'd be very grateful. 3nodding
EDIT: I should just probably point out here that by doing further maths, you're assumed to know last year's work of maths; as I'm only in the second last year this can make fine points difficult as I haven't been taught the last year of maths yet. If anyone with experience like this can tell me how they coped, I'd be equally as grateful.
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Posted: Wed May 16, 2007 6:00 am
with integration, it's just mechanical work, so once you've done the thought processes behind understanding it, everything is summarized on nice tables. you just have to look it up. That's what we do for our physics assignments, some integration can get messy.
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Posted: Wed May 16, 2007 8:52 am
poweroutage with integration, it's just mechanical work, so once you've done the thought processes behind understanding it, everything is summarized on nice tables. you just have to look it up. That's what we do for our physics assignments, some integration can get messy. I know, but there are some functions that aren't in tables that we're assumed to learn regardless of whether we've done A2 maths or not. I know integration can get messy; but thanks for that. I'll have a look through the books again to see if there's any essentical functions I ought to know.
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Posted: Wed May 16, 2007 6:02 pm
Slartibartfarst poweroutage with integration, it's just mechanical work, so once you've done the thought processes behind understanding it, everything is summarized on nice tables. you just have to look it up. That's what we do for our physics assignments, some integration can get messy. I know, but there are some functions that aren't in tables that we're assumed to learn regardless of whether we've done A2 maths or not. I know integration can get messy; but thanks for that. I'll have a look through the books again to see if there's any essentical functions I ought to know. in which case you have the integrator: wolfram.integrator.com or just google integrator
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Posted: Fri May 18, 2007 2:42 pm
poweroutage Slartibartfarst poweroutage with integration, it's just mechanical work, so once you've done the thought processes behind understanding it, everything is summarized on nice tables. you just have to look it up. That's what we do for our physics assignments, some integration can get messy. I know, but there are some functions that aren't in tables that we're assumed to learn regardless of whether we've done A2 maths or not. I know integration can get messy; but thanks for that. I'll have a look through the books again to see if there's any essentical functions I ought to know. in which case you have the integrator: wolfram.integrator.com or just google integrator I found them before this post and book marked them, but thanks anyway. 3nodding
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Posted: Sat May 19, 2007 6:59 am
so I guess your question is about how to integrate. How do they remember? They don't, it's all written down somewhere and if they want to look it up in a book they can. But no one bothers doing the mechanics of integration these days. Does that answer your question?
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Posted: Sun May 20, 2007 11:53 pm
As far as I'm concerned, if I actually have to do the integrations out, there are three main things to remember:
int n*x^(n-1) = x^n polynomials, except when n = 0, in which case int x^(-1) = ln(x)
int a*e^(ax) = e^(ax) exponentials
int f(x)*g'(x) = f(x)*g(x) - int f'(x)*g(x) integration by parts
The first one should be fairly easy to recognize, the second as well. The third is for any time x shows up more than once in a term.
Then we just remember that cos(ax) = (e^(iax)+e^-iax)/2, sin(ax) = (e^(iax)-e^(-iax))/2, which gives us that int a*cos(ax) = sin(ax) and int a*sin(ax) = -cos(ax)
There are also some other things to look out for, like powers of 1+x^2 in the denominator, which means an inverse trig function, and the evil e^(-ax^2), which can't really be integrated well except from 0 to infinity or from -infinity to infinity.
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Posted: Mon May 21, 2007 8:56 am
Layra-chan As far as I'm concerned, if I actually have to do the integrations out, there are three main things to remember: int n*x^(n-1) = x^n polynomials, except when n = 0, in which case int x^(-1) = ln(x) int a*e^(ax) = e^(ax) exponentials int f(x)*g'(x) = f(x)*g(x) - int f'(x)*g(x) integration by parts The first one should be fairly easy to recognize, the second as well. The third is for any time x shows up more than once in a term. Then we just remember that cos(ax) = (e^(iax)+e^-iax)/2, sin(ax) = (e^(iax)-e^(-iax))/2, which gives us that int a*cos(ax) = sin(ax) and int a*sin(ax) = -cos(ax) There are also some other things to look out for, like powers of 1+x^2 in the denominator, which means an inverse trig function, and the evil e^(-ax^2), which can't really be integrated well except from 0 to infinity or from -infinity to infinity. Thanks. I've looked up all I needed and you reminded me of what I did.
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Posted: Tue May 22, 2007 4:51 am
Layra-chan As far as I'm concerned, if I actually have to do the integrations out, there are three main things to remember: int n*x^(n-1) = x^n polynomials, except when n = 0, in which case int x^(-1) = ln(x) int a*e^(ax) = e^(ax) exponentials int f(x)*g'(x) = f(x)*g(x) - int f'(x)*g(x) integration by parts The first one should be fairly easy to recognize, the second as well. The third is for any time x shows up more than once in a term. Then we just remember that cos(ax) = (e^(iax)+e^-iax)/2, sin(ax) = (e^(iax)-e^(-iax))/2, which gives us that int a*cos(ax) = sin(ax) and int a*sin(ax) = -cos(ax) There are also some other things to look out for, like powers of 1+x^2 in the denominator, which means an inverse trig function, and the evil e^(-ax^2), which can't really be integrated well except from 0 to infinity or from -infinity to infinity. hm, what class do you do it for? Do they still test your integration abilities as a subtopic of a question on exams?
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Posted: Thu May 24, 2007 8:23 pm
poweroutage Layra-chan As far as I'm concerned, if I actually have to do the integrations out, there are three main things to remember: int n*x^(n-1) = x^n polynomials, except when n = 0, in which case int x^(-1) = ln(x) int a*e^(ax) = e^(ax) exponentials int f(x)*g'(x) = f(x)*g(x) - int f'(x)*g(x) integration by parts The first one should be fairly easy to recognize, the second as well. The third is for any time x shows up more than once in a term. Then we just remember that cos(ax) = (e^(iax)+e^-iax)/2, sin(ax) = (e^(iax)-e^(-iax))/2, which gives us that int a*cos(ax) = sin(ax) and int a*sin(ax) = -cos(ax) There are also some other things to look out for, like powers of 1+x^2 in the denominator, which means an inverse trig function, and the evil e^(-ax^2), which can't really be integrated well except from 0 to infinity or from -infinity to infinity. hm, what class do you do it for? Do they still test your integration abilities as a subtopic of a question on exams? The only classes I have that still require actual integration are my physics classes; my math classes just assume that something is integrable and don't really care what they turn out to be.
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Posted: Fri May 25, 2007 12:23 pm
Layra-chan The only classes I have that still require actual integration are my physics classes; my math classes just assume that something is integrable and don't really care what they turn out to be. I love that feeling. "I can prove that this is integrable. Let the scientists find out what the integral actually is."
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Posted: Fri May 25, 2007 10:28 pm
Swordmaster Dragon Layra-chan The only classes I have that still require actual integration are my physics classes; my math classes just assume that something is integrable and don't really care what they turn out to be. I love that feeling. "I can prove that this is integrable. Let the scientists find out what the integral actually is." I hate that part. crying
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Posted: Sat May 26, 2007 7:27 am
Integration can be made a lot easier if you just expand the function, alter it, and collapse it (then you don't even need to know how to integrate sin(x)). Also, just memorize your inverse trig integrations, and the very basics (integration by parts, substitution, 1/x) and you should be fine.
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Posted: Thu Jul 12, 2007 10:02 pm
With rules do you mean knowing when to use the chain rules, product rule and quotient rule?
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Posted: Thu Jul 12, 2007 10:08 pm
Layra-chan As far as I'm concerned, if I actually have to do the integrations out, there are three main things to remember: int n*x^(n-1) = x^n polynomials, except when n = 0, in which case int x^(-1) = ln(x) int a*e^(ax) = e^(ax) exponentials int f(x)*g'(x) = f(x)*g(x) - int f'(x)*g(x) integration by parts The first one should be fairly easy to recognize, the second as well. The third is for any time x shows up more than once in a term. Then we just remember that cos(ax) = (e^(iax)+e^-iax)/2, sin(ax) = (e^(iax)-e^(-iax))/2, which gives us that int a*cos(ax) = sin(ax) and int a*sin(ax) = -cos(ax) There are also some other things to look out for, like powers of 1+x^2 in the denominator, which means an inverse trig function, and the evil e^(-ax^2), which can't really be integrated well except from 0 to infinity or from -infinity to infinity. As for e^(-ax^2), Ramanujan figured out how to integrate it in the form of a continued fraction. I'll edit my post tomorrow to include it.
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