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Question

The value of the integral π/22/πtan1xxdx is 
  1. π4ln(π2) 
  2. π2ln(π2) 
  3. πln(π2) 
  4. 12ln(π2) 


Solution

The correct option is B π2ln(π2) 
Let I=π/22/πtan1xxdx     (1)
Put x=1tdx=1t2dt
I=2/ππ/2tcot1t(1t2dt)
=2/ππ/2cot1ttdt
I=π/22/πcot1ttdt=π/22/πcot1xxdx     (2)

Adding (1) and (2), we get
2I=π/22/ππ/2xdx
I=π4lnπ24=π2ln(π2)

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