If `I_(1)=int_(0)^(1)(dx)/(e^(x)(1+x))` and `I_(2)=int_(0)^(pi//4)(e^(tan^(2)theta)sintheta)/((2-tan^(2)theta)cos^(3)theta) d theta`,then find the value of `(l_(1))/(l_(2))`.

Answer

`2//e`

Explanation

`I_(2)int_(0)^(pi//4)(e^(tan^(2)theta).tan theta)/((2-tan^(2)theta))sec^(2) d theta`
Put `tan^(2)theta=t`
`:.2 tan theta sec^(2) d theta =dt`
`:.I_(2)=1/2int_(0)^(1)(e^(t)dt)/((2-t))`
`=1/2 int_(0)^(1)(e^(1-t)dt)/(1+t)`
`=e/2int_(0)^(1)(dt)/(e^(t)(t+1))=e/2.I_(1)`
`:. (I_(1))/(I_(2))=2/e`