The value of `int_(1/e)^(tanx)(tdt)/(1+t^2)+int_(1/e)^(cotx)(dt)/(t(1+t^2)),` where `x in (pi/6,pi/3)` , is equal to: 0 (b) 2 (c) 1 (d) none of these

Options

A. 0
B. 2
C. 1
D. none of these

Answer

C

Explanation

`f(x)=int_(1/e)^(tanx)(tdt)/((1+t^(2)))+int_(1/e)^(cotx)(dt)/(t(1+t^(2)))`
`:.f'(x)=(tanx)/(1+tan^(2)x)"sec"^(2)x+1/(cotx(1+cot^(2)x))(-cosec^(2)x)`
`=tanx-tanx=0`
Thus, `f(x)` is a constant function.
`f((pi)/4)=int_(1/e)^(1)(tdt)/((1+t^(2)))+int_(1/e)^(1)(dt)(t(1+t^(2)))`
`=int_(1/e)^(1)1/t dt=int_(1//e)^(1)=1`