1.

Let l_1=int_0^ooln(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»

`l_1 GT l_2`
`l_2 gt l_1`
`l_1=l_2`
`l_1+l_2=0`

Solution :`I_2=int_0^(pi//2)(theta/(sin theta))^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
`=-2 int_0^(pi//2) ln sin theta =piln2`
`I_1=int_0^oo ln (x+1/x)(dx)/(1+x^2)`
Let `tan^(-1) x = theta`
`=int_0^(pi//2) ln (tan theta+cot theta)d theta`
`=-int_0^(pi//2) (ln sin theta + ln cos theta ) d theta =pi ln2`
`THEREFORE I_1=I_2`


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