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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Evaluate the definite integral\[\int\limits_0^\frac{\Large \pi}{\Large 4}\tan x\;dx\]

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Toolbox:
  • (i)$ \int \limits_a^b f(x)dx=F(b)- F(a)$
  • (ii)$\int \tan x dx= -log |\cos x| \;or\; log|\sec x| $
Given $ I=\int \limits_0^{\pi/4} \tan x dx$
 
On integrating we get
 
$\bigg[log |sec x|\bigg]^{\pi/4}_0$
 
On applying limits we get
 
$log(\sec \frac{\pi}{4}-\sec 0)$
 
But $\sec \frac{\pi}{4}=\sqrt 2\;and\;\sec 0=1$
 
Therefore $I=log|\sqrt 2-1|$
 
But $ log a-log b=log(a/b),$ similarly
 
$log \frac{\sqrt 2}{1}=log \sqrt 2$
 
$=log 2^{1/2}$
 
But $ log x^a=a log x,$ similarly,
 
We can write $\log 2^{1/2}=\frac{1}{2} log 2$
 
Hence $\int \limits_0^{\pi/4} \tan x dx=\frac{1}{2} log 2$

 

answered Feb 10, 2013 by meena.p
 
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