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the value of int(1/e->tanx) (tdt)/(1+t^2...

the value of `int_(1/e->tanx) (tdt)/(1+t^2) + int_(1/e->cotx) (dt)/(t*(1+t^2)) =`

A

0

B

2

C

1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`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`
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