1.

Let I_(1)=int_(0)^(oo)ln(x+1/x)(dx)/(1+x^(2)) and l_(2)=int_(0)^(pi//2)((theta)/(sintheta))^(2)d theta, then which of the following is/are correct?

Answer»

`I_(1)gtI_(2)`
`I_(2)gtI_(1)`
`I_(1)=I_(2)`
`I_(1)+I_(2)=0`

SOLUTION :`I_(2)=int_(0)^(pi//2)((theta)/(sintheta))^(2) d theta=int_(0)^(pi//2) theta^(2)cosec^(2) theta d theta`
`=[-theta^(2) cot theta]_(0)^(pi//2)+int_(0)^(pi//2) 2 theta cot theta d theta`
Use: integration by parts
`=- int_(0)^(pi//2)L nsin theta=pi l n 2`
`I_(1)=int_(0)^(oo)ln(X+1/x)(dx)/(1+x^(2))`
Let `tan^(-1)x = theta`
`=int_(0)^(pi//2)l n (tan theta+cot theta) d theta`
`=-int_(0)^(pi//2)(ln sin theta+l n cos theta) d theta=pi l N2`
`:.I_(1)=I_(2)`


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