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Integration advice needed (2 Viewers)
- Thread starter leehuan
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Paradoxica
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Use integration by parts to construct a two-step recurrence relation in both variables.
Alternatively, convert it into the fundamental trigonometric ratios and reduce the cosines and sines.
Alternatively, convert it into the fundamental trigonometric ratios and reduce the cosines and sines.
leehuan
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Oh dear. So there isn't a trivial substitution like the two other cases have?
leehuan
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Hm. What's an example integral where l is even?
What they mean by equations 1 is the circle identity of secx and tanx, and the derivative of tanx and secx.
leehuan
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Wait... Did I make a stupid in my WolframAlpha?^2$. So $I = \int \sec ^5 x \left(\sec^2 x-1\right)^2\text{ d}x$. Expanding this via the binomial theorem will give us some terms with an even (and some with odd) power of $\sec x$.$)
http://www.wolframalpha.com/input/?i=expand+sec^5x(sec^2x-1)^2
leehuan
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Ouch. Figures, my suspicions were right. (Might need a tiny revision in those notes @ photastic)Oh sorry lol, I was thinking of a sec rather than sec^2 being the thing expanded.
^k = \left(\sec^2 x -1\right)^k$. Powers of $\sec x$ in this expansion will all be even and then multiplying it with the $\sec ^m x$, they'll all become odd as $m$ is odd. So in the end, all the secant powers will be odd powers, which can be tediously computed using a secant reduction formula method for instance.$)
Thank goodness we have reduction formulae I guess...
Yeah, you may want to pre-derive that formula before setting out to tackle an integral like this so that you have it handy if you need it.
Paradoxica
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As an aside, what about the ever so disgusting tangent half angle substitution?
Is that a viable method?
Is that a viable method?
Lol they're from MATH1251
leehuan
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Actually the Ext 2 syllabus specifically states that these questions are NOT part of the course:How would you tackle integrals of the form
where m is odd and n is even?
(Within the scope of MX2)
"Recurrence relations such as ....... which involve more than one integer parameter, are excluded."
Maybe leehuan meant when the m and n are particular numerical values rather than parameters? E.g. finding something like ∫sec3 x tan2 x dx.
Paradoxica
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I see no reason as to why double parameter integrals should not be part of the HSC.
Actually he probably did. I didn't read the post properly and missed "of the form".