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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Evaluate the definite integral\[\int\limits_\frac{\Large \pi}{\Large 6}^\frac{\Large \pi}{\Large 4}cosec x\;dx\]

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Toolbox:
  • $ \int \limits_a^b f(x)dx=F(b)- F(a)$
  • (ii)$\int cosec x dx= log |cosec x- \cot x|+c $
Given $ I=\int \limits_{\pi/6}^{\pi/4} cosec x dx$
 
On integrating we get
 
$\bigg[log |cosec x -\cot |\bigg]^{\pi/4}_\frac{\pi}{6}$
 
On applying limits we get
 
$ I=log \bigg[(cosec \frac{\pi}{4}- \cot \frac{\pi}{4})-(cosec \frac{\pi}{6}-\cot \frac{\pi}{6})\bigg]$
 
$ I=log[(\sqrt 2-1)-(2-\sqrt 3)]$
 
$log a-log b=log(a/b),$ similarly,
 
$I=log \bigg[\large\frac{(\sqrt 2 -1)}{2- \sqrt 3}\bigg]$

 

answered Feb 10, 2013 by meena.p
 
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