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int(0)^(pi//2)(1)/((1+tanx))dx=?...

`int_(0)^(pi//2)(1)/((1+tanx))dx=?`

A

`pi`

B

`pi//2`

C

`pi//3`

D

`pi//4`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `l = int_(0)^(pi//2)(dx)/(1+tan x)`
`l = int_(0)^(pi//2) (cos x)/(sin x + cos x)` ….(i)
`l = int_(0)^(pi//2)(cos((pi)/(2)-x))/(sin((pi)/(2)-x)+cos((pi)/(2)-x))dx`
`= int_(0)^(pi//2)(sin x)/(cos x+ sin x)dx` ….(ii)
On adding Eqs. (i) and (ii), we get
`2 l = int_(0)^(pi//2)((sin x + cos x)/(sin x + cos x))dx`
`= int_(0)^(pi//2)dx = (pi)/(2)rarr l = (pi)/(4)`
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