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Integral Calculus

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29 Jul 2009 18:21:09 IST

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Hello there, i am trying to integrate (2secx)/(2+tanx).i have been informed that it is hyperbolic f

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Hello there, i am trying to integrate (2secx)/(2+tanx).i have been informed that it is hyperbolic function but i havent covered that yet in the syllabus.I can express sec as 1/cosx and tan is sinx/cosx.I can cancel it down to 2(1/sinx)+2 but dont know whether that is allowed.It is part of a definate integral with the limits PI/3 and -PI/4.Anyone help???

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Anant Kumar

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Joined: 10 Jul 2008

Posts: 598

5 Aug 2009 08:31:36 IST

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If the integral you gave is
$I=\int_{-\pi/4}^{\pi/3} \dfrac{2\sec x}{2+\tan x}\ \mathrm dx$ ,
then its not hyperbolic or anything like that.
One has
$\dfrac{2\sec x}{2+\tan x} = \dfrac{2}{2\cos x+\sin x}=\dfrac{2}{\sqrt{5}\sin(x+\alpha)}$
where $\tan \alpha =2$
Therefore,
$I=\int_{-\pi/4}^{\pi/3} \dfrac{2}{\sqrt{5}\sin(x+\alpha)}\ \mathrm dx=\dfrac{2}{\sqrt{5}}\left|\ln\tan\frac{x}{2}\right|_{-\p...$

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