Cotangent function integration problem

[Benny]

I was doing an integration question earlier on and I came across something that I would like to be cleared up. The question basically boiled down to:

$$- \left[ {\cot \left( \theta \right) + \theta } \right]_{\frac{\pi }{6}}^{\frac{\pi }{2}} = - \left[ {\frac{{\cos \left( \theta \right)}}{{\sin \left( \theta \right)}} + \theta } \right]_{\frac{\pi }{6}}^{\frac{\pi }{2}}$$

Now if I just substitute the relevant values into the antiderivative it works out fine. However if I write the following I end up getting a 'weird' (I do not
know the right words to describe it ) answer(something involving 1/infinity).

$$- \left[ {\cot \left( \theta \right) + \theta } \right]_{\frac{\pi }{6}}^{\frac{\pi }{2}} = - \left[ {\frac{1}{{\tan \left( \theta \right)}} + \theta } \right]_{\frac{\pi }{6}}^{\frac{\pi }{2}}$$

Can someone explain to me why this occurs? I can understand that when certain values are substituted in, 'weird' numbers appear but I cannot understand why the question works/does not work, depending on which way an expression is written(I am referring to the cotangent function), even though it is just the same thing. Any help would be good. I hope I was not unclear.

 

[dextercioby]

That's because [itex] \frac{\pi}{2} [/tex] is the domain of cotangent (cot pi/2=0) and not in the domain of tangent (tan pi/2 does not exist).Taking this int consideration,at best u can do is set a limit for the 1/tangent.But again,why do that,when pi/2 is clearly in the domain on cotangent and the result works out fine?

Daniel.

 

[Benny]

Thanks for clearing that up for me.