Robert G.

asked • 12/22/19

What happens when the given integral of a function has domain restrictions.

The integral of cosec x dx is given as ln (cosec x - cot x) yet this function has domain restrictions eg at x = 5. This seems to be the tip of the iceberg on integration.


The given integral of arcoth x is given as x arcoth x + 1/2 ln (1-x^2) - and some variations of this - but this is undefined for x real. No-one ever seems to comment on this or even be aware of it.


On working through the integration of the 24 trig functions this problem appears to crop up repeatedly but in 50 years of mathematices I'd never noticed it before.


What am I not understanding? Is it just that there is no simple expression for the indefinite integral of cosec 5 and more significantly no expression at all for the integral of arcoth whatever the websites might say.

Mark M.

The indefinite integral of cosecant x is -ln |csc x + cot x| + C. That may be the crux of the matter.
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12/22/19

Robert G.

Fair point but take the derivative of arccosec given as -1/x(1-x^2)^0.5. The function domain and the derivative domain have no overlap so the derivative is completely invalid. What am I not understanding?
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12/23/19

Adam M.

How do you figure there is no overlap? The domain of arccosec is (-\infty, -1] U [1,\infty). It is the anti derivative of -1/x(1-x^2)^0.5 on the domain (-\infty, -1) U (1,\infty). At -1 and 1, arccosec is -\pi/2 or \pi/2, and it's derivative is undefined. The function f(x)=-1/x(1-x^2)^0.5 on the other hand has domain (-\infty,-1) U (-1,1) U (1,\infty). As mentioned above, on (-\infty, -1) U (1,\infty), f(x) is the derivative of arccosec. On (-1,1), it just isn't (because arccosec isn't even defined there).
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12/24/19

Robert G.

Thank you Mark and Adam for your helpful comments. I have revisited this whole issue. I mapped out all 72 domains (diff. and int. of all 24 trig functions) and have indeed got them all to line up by inserting absolute values where necessary. For example on integral arccosec given as x.arccosec x + arcosh x the second term needs to be arcosh abs x. As Adam says you get a few isolated undefined points which can be discounted. Now I can sleep easy.
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12/24/19

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Mark H. answered • 12/27/19

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I'll start only by not understanding the 1st one.....csc(x) "blows up" at x = n*pi, therefor, you can not integrate through those values.

"domain restriction at x = 5"---what does that mean? csc(x) has minima at x = n*pi/2, but the domain restrictions would be where the function blows up.

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