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Recursion Formula (1 Viewer)
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alakazimmy
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[cosec(x)]^2 dx = - d (cot(x))
Using that relationship, you can form your recursion formula, with integration by parts.
Hope this helps.
Using that relationship, you can form your recursion formula, with integration by parts.
Hope this helps.
sorry it was meant to be [cosec(x)]^n
L
quit bos forever 23/01/07
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Timothy.Siu
Prophet 9
change (cosec x)^n to (cosec x)^(n-2) . cosec^2 x
and do it by parts,
then in the integration part, change cot^2 x to cosec^2 x-1
and ur done!
and do it by parts,
then in the integration part, change cot^2 x to cosec^2 x-1
and ur done!
alakazimmy
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The same relation will still apply.
You pull out a [cosec(x)]^2, so you'll be left with [cosec(x)]^(n-2) * d (cot(x))
Then integration by parts will solve all
I did exactly what you did originally but my problem is cot(pi/2)=1/tan(pi/2) undefined?\\ \\ =\ \ 2^{n-2}\sqrt{3}\ \ - \ (n-2)\int_{\frac{\pi }{6}}^{\frac{\pi }{2}}\ cot^{2}x\ csc^{n-2}x\ dx\\ \\ =\ \ 2^{n-2}\sqrt{3}\ -\ (n-2) \int_{\frac{\pi }{6}}^{\frac{\pi }{2}}\left (csc^{n} x\ -\ csc^{n-2}x\right )\ dx\\ \\ =\ \ 2^{n-2}\sqrt{3}\ -\ (n-2)I_{n}\ +\ (n-2)I_{n-2}\\ \\ \therefore \ \ (n-1) I_{n}\ \ =\ \ 2^{n-2}\sqrt{3}\ \ +\ \ (n-2)I_{n-2}\\ \\ etc\ etc)
alakazimmy
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cot(pi/2) = 0, as limit of tan(x) as x approaches pi/2 is +infinity
cool. that's the only problem i had with it.
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The trig function cot x by definition is cos x / sin x. So when you sub π/2
cot π/2 = cos π/2 / sin π/2 = 0 since cos π/2 = 0.
tan x = sin x / cos x by definition
hence
cot x = 1 / tan x
ONLY when sin x and cos x are non-zero.
Strictly speaking, the cotangent function is defined as,
cot x = cos x / sin x
which requires sin x being non-zero, but can exist for cos x being zero. (In other words, cot x =/= 1/tan x if cos x = 0)
cot π/2 = cos π/2 / sin π/2 = 0 since cos π/2 = 0.
tan x = sin x / cos x by definition
hence
cot x = 1 / tan x
ONLY when sin x and cos x are non-zero.
Strictly speaking, the cotangent function is defined as,
cot x = cos x / sin x
which requires sin x being non-zero, but can exist for cos x being zero. (In other words, cot x =/= 1/tan x if cos x = 0)
ok.