Another Hard Integration Question (1 Viewer)
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Pwnage101
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decompose into partial fractions
You'll need to use complex numbers to factorise the denominator into four complex factors (by using de moivres theorm and solving (u^4)-1 = 0), which then reduces to two real quadratics when you multiply two of these out, which you then decompose into partial fractions and integrate normally, you will end up with terms in log and inverse tan (2 of each).
Takes a bit of work though.
Also since this is a definite integral, you should integrate from 1 to t, where t is a dummy variable, because you then take the limit as t tends to infinity.
A bit beyond HSC though.
You'll need to use complex numbers to factorise the denominator into four complex factors (by using de moivres theorm and solving (u^4)-1 = 0), which then reduces to two real quadratics when you multiply two of these out, which you then decompose into partial fractions and integrate normally, you will end up with terms in log and inverse tan (2 of each).
Takes a bit of work though.
Also since this is a definite integral, you should integrate from 1 to t, where t is a dummy variable, because you then take the limit as t tends to infinity.
A bit beyond HSC though.
Last edited:
Top Secret
Member
What course are you doing at uni?
GUSSSSSSSSSSSSS
Active Member
even the esteemed drongoski had to "tip his hat" to this i guess =P
Pwnage101
Moderator
Like i said, the question as is is beyond the HSC. Thats not to say the concepts that you need are beyond HSC - they are not, however, the only way they could use this in an exam would be to guide you.
Eg
part a: solve u^4-1=0
part b: hence factorise u^4-1 into the product of two real quadratics
part c: hence decompose (1+u^4)^-1 into partial fractions
part d:hence find the indefinite integral
part e: integrate from 1 to t
part f: what can we say about the integral as t tends to infinity
But i highly doubt it would ever be asked, even like that. However it is a great question to test how well you can combine several topics in the MX2 course, with a bit of guidance.
Eg
part a: solve u^4-1=0
part b: hence factorise u^4-1 into the product of two real quadratics
part c: hence decompose (1+u^4)^-1 into partial fractions
part d:hence find the indefinite integral
part e: integrate from 1 to t
part f: what can we say about the integral as t tends to infinity
But i highly doubt it would ever be asked, even like that. However it is a great question to test how well you can combine several topics in the MX2 course, with a bit of guidance.
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jet
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Yeah, I just realised. Fixing it now.what's that thing you did with the inverse tans? :S
should be:))
Pwnage101
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well done jetblack2007.
Aerath
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Yeah, that'd be a great question, in my opinion.
q3thefish
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Extremely impressive.Pretty sure I've made a numerical error somewhere at the end - I get twice the value that my grapher gets. I'll check working tomorrow. The method should be correct though. Edit: Fixed problem thanks to lolokay.
