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int(pi//6)^(pi//3)(1)/((1+sqrt(tanx)))dx...

`int_(pi//6)^(pi//3)(1)/((1+sqrt(tanx)))dx=(pi)/(12)`

A

`pi/12`

B

`pi/2`

C

`pi/6`

D

`pi/4`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `l=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tan x))= int_(pi//6)^(pi//3)((sqrt(cos x))/(sqrt(sin x)+sqrt(cos x)))dx`
`rArr l = int_(pi//6)^(pi//3)(sqrt(sin x))/(sqrt(cos x)+sqrt(sin x))dx` ….(ii)
`[because int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx]`
On adding Eqs. (i) and (ii), we get
`2l=int_(pi//6)^(pi//3)1dx=[x]_(pi//6)^(pi//3)=(pi)/(6)`
`rArr " " l = (pi)/(12)`
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